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Subbase

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about an object in mathematical topology. For the term in highway engineering, see <a href="Subbase_(pavement)" title="Subbase (pavement)">Subbase (pavement)</a>. For the frequency range, see <a href="Sub-bass" title="Sub-bass">Sub-bass</a>.</div>

In <a href="Topology" title="Topology">topology</a>, a subbase (or subbasis, prebase, prebasis) for the <a href="Topology_(structure)" class="mw-redirect" title="Topology (structure)">topology</a> τ of a <a href="Topological_space" title="Topological space">topological space</a> (X, τ) is a subcollection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> that generates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ,}">

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<annotation encoding="application/x-tex">{\displaystyle \tau ,}</annotation>

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</math><img src="./26d6cc28c28ff4ff88402f47f2a99e583e9e045f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.849ex; height:2.009ex;" alt="{\displaystyle \tau ,}" loading="lazy"> in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is the smallest topology containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

Subbase is a weaker notion than that of a <a href="Base_(topology)" title="Base (topology)">base</a> for a topology.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a topological space with topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

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<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

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</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"> A subbase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is usually defined as a subcollection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> satisfying one of the three following equivalent conditions:

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is the smallest topology containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">: any topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ^{\prime }}">

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<annotation encoding="application/x-tex">{\displaystyle \tau ^{\prime }}</annotation>

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</math><img src="./1d4eea27fc77853645e3c34740a8573959a35edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.943ex; height:2.509ex;" alt="{\displaystyle \tau ^{\prime }}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> must also contain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

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<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

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</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is the <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of all topologies on X containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}">

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<annotation encoding="application/x-tex">{\displaystyle B.}</annotation>

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</math><img src="./0eccf5bca7cdc1fa4439af2d31831db6bde00473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}" loading="lazy"></li>

<li>The collection of open sets consisting of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and all finite <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersections</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> forms a <a href="Basis_(topology)" class="mw-redirect" title="Basis (topology)">basis</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

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<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

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</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"><a href="#cite_note-FOOTNOTERudin1991392_Appendix_A2-1">[1]</a><a href="#cite_note-2">[note 1]</a> This means that every proper <a href="Open_set" title="Open set">open set</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> can be written as a <a href="Union_(set_theory)" title="Union (set theory)">union</a> of finite intersections of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}">

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<annotation encoding="application/x-tex">{\displaystyle B.}</annotation>

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</math><img src="./0eccf5bca7cdc1fa4439af2d31831db6bde00473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}" loading="lazy"> Explicitly, given a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subsetneq X,}">

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<annotation encoding="application/x-tex">{\displaystyle U\subsetneq X,}</annotation>

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</math><img src="./bdc1b2d3c01bf6ef8cf33d7adb1eff011e6b028e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.508ex; height:2.676ex;" alt="{\displaystyle U\subsetneq X,}" loading="lazy"> there are finitely many sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1},\ldots ,S_{n}}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle S_{1},\ldots ,S_{n}}</annotation>

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</math><img src="./592d6aadf39f9996a9a0431ca320151d74643b7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.301ex; height:2.509ex;" alt="{\displaystyle S_{1},\ldots ,S_{n}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle B,}</annotation>

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</math><img src="./075d661417b8ca5a991a2a7bd4991cc1ab856d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle B,}" loading="lazy"> such that the intersection of these sets contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

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<annotation encoding="application/x-tex">{\displaystyle U.}</annotation>

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</math><img src="./7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy"></li></ol>

If we additionally assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> <a href="Cover_(topology)" title="Cover (topology)">covers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, or if we use the <a href="Nullary_intersection" class="mw-redirect" title="Nullary intersection">nullary intersection</a> convention, then there is no need to include <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in the third definition.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is a subbase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> generates the topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

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</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"> This terminology originates from the explicit construction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> using the second or third definition above.

Elements of subbase are called subbasic (open) sets. A <a href="Cover_(topology)" title="Cover (topology)">cover</a> composed of subbasic sets is called a subbasic (open) cover.

For any subcollection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> of the <a href="Power_set" title="Power set">power set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (X),}">

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<annotation encoding="application/x-tex">{\displaystyle \wp (X),}</annotation>

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</math><img src="./09d65fa8f234b64d1e2a6c4b2255bbaef55f059d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.914ex; height:2.843ex;" alt="{\displaystyle \wp (X),}" loading="lazy"> there is a unique topology having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> as a subbase; it is the intersection of all topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. In general, however, the converse is not true, i.e. there is no unique subbasis for a given topology.

Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (X)}</annotation>

</semantics>

</math><img src="./bf7050c843936eee5cb9985efbe0d2289f747e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.843ex;" alt="{\displaystyle \wp (X)}" loading="lazy"> and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the three conditions is more useful than the others.

<div class="mw-heading mw-heading3"><h3 id="Alternative_definition">Alternative definition</h3></div>

Less commonly, a slightly different definition of subbase is given which requires that the subbase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"><a href="#cite_note-FOOTNOTEMunkres200082-3">[2]</a> In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the union of all sets contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}.}</annotation>

</semantics>

</math><img src="./33eb5608b7d9cdddb144c0b6e00ded582237ea27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.19ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}.}" loading="lazy"> This means that there can be no confusion regarding the use of nullary intersections in the definition.

However, this definition is not always equivalent to the three definitions above. There exist topological spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> with subcollections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}\subseteq \tau }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}\subseteq \tau }</annotation>

</semantics>

</math><img src="./574784693c5092c499e9c462202d87ab913127d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.844ex; height:2.343ex;" alt="{\displaystyle {\mathcal {B}}\subseteq \tau }" loading="lazy"> of the topology such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is the smallest topology containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy">, yet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> does not cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. For example, consider a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =\{\varnothing ,\{p\},X\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =\{\varnothing ,\{p\},X\}}</annotation>

</semantics>

</math><img src="./a2bd5bc1a3af1525cdefb9390f37f380a7ea26ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.976ex; height:2.843ex;" alt="{\displaystyle \tau =\{\varnothing ,\{p\},X\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}=\{\{p\}\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}=\{\{p\}\}}</annotation>

</semantics>

</math><img src="./c286a7e8d477cc7aa2172c5ad47263834f9e9709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.461ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}=\{\{p\}\}}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in X.}</annotation>

</semantics>

</math><img src="./180e351f0e4c54e950e0cb08086ccd789e0d76b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.726ex; height:2.509ex;" alt="{\displaystyle p\in X.}" loading="lazy"> Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> is a subbase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">, yet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> doesn't cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> as long as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>

</semantics>

</math><img src="./901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy"> elements. In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies the <a href="T1_separation_axiom" class="mw-redirect" title="T1 separation axiom">T1 separation axiom</a> must be a cover of that space.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

The topology generated by any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}\subseteq \{\varnothing ,X\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}\subseteq \{\varnothing ,X\}}</annotation>

</semantics>

</math><img src="./affc51f60be0103bde14671db9ca4faad2ff1f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.738ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}\subseteq \{\varnothing ,X\}}" loading="lazy"> (including by the empty set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}:=\varnothing }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}:=\varnothing }</annotation>

</semantics>

</math><img src="./213b1e493dbc5ec282c91a7b4897c4c325be10a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.046ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}:=\varnothing }" loading="lazy">) is equal to the trivial topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varnothing ,X\}.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\varnothing ,X\}.}</annotation>

</semantics>

</math><img src="./5f49b1b47a4a792c593f92d4f2da450440590bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.794ex; height:2.843ex;" alt="{\displaystyle \{\varnothing ,X\}.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> is a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> then the topology generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

</semantics>

</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"> Thus any basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> for a topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is also a subbasis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

</semantics>

</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> is any subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> then the topology generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> will be a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

</semantics>

</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy">

The usual topology on the <a href="Real_number" title="Real number">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> has a subbase consisting of all <a href="Semi-infinite" title="Semi-infinite">semi-infinite</a> open intervals either of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-\infty ,a)}</annotation>

</semantics>

</math><img src="./c95e49fd3f33ee3f4bb536214d7860f0eab27577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle (-\infty ,a)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b,\infty ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b,\infty ),}</annotation>

</semantics>

</math><img src="./ed479b83602944723a8de4179d368babcfe71994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.811ex; height:2.843ex;" alt="{\displaystyle (b,\infty ),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are real numbers. Together, these generate the usual topology, since the intersections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)=(-\infty ,b)\cap (a,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)=(-\infty ,b)\cap (a,\infty )}</annotation>

</semantics>

</math><img src="./a73c9b81eb1d23dc5409d5f4364e91082d9810c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.121ex; height:2.843ex;" alt="{\displaystyle (a,b)=(-\infty ,b)\cap (a,\infty )}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation>

</semantics>

</math><img src="./41558abc50886fdf38817495b243958d7b3dd39b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}" loading="lazy"> generate the usual topology. A second subbase is formed by taking the subfamily where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are <a href="Rational_number" title="Rational number">rational</a>. The second subbase generates the usual topology as well, since the open intervals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation>

</semantics>

</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,}</annotation>

</semantics>

</math><img src="./f059f053fcf9f421b7c74362cf3bd5ed024e19d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> rational, are a basis for the usual Euclidean topology.

The subbase consisting of all semi-infinite open intervals of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-\infty ,a)}</annotation>

</semantics>

</math><img src="./c95e49fd3f33ee3f4bb536214d7860f0eab27577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle (-\infty ,a)}" loading="lazy"> alone, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is a real number, does not generate the usual topology. The resulting topology does not satisfy the <a href="T1_space" title="T1 space">T1 separation axiom</a>, since if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a<b}</annotation>

</semantics>

</math><img src="./91a7698e4c7401bb321f97888b872b583a9e4642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}" loading="lazy"> every <a href="Open_set" title="Open set">open set</a> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> also contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a.}</annotation>

</semantics>

</math><img src="./6b803da9c45c1186883bde55107e9ccb102c92c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}" loading="lazy">

The <a href="Initial_topology" title="Initial topology">initial topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> defined by a family of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}:X\to Y_{i},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}:X\to Y_{i},}</annotation>

</semantics>

</math><img src="./bf732943bb677f470954c32cd51e06cd5af8fc1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.267ex; height:2.509ex;" alt="{\displaystyle f_{i}:X\to Y_{i},}" loading="lazy"> where each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation>

</semantics>

</math><img src="./d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" loading="lazy"> has a topology, is the coarsest topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> such that each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation>

</semantics>

</math><img src="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"> is <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a>. Because continuity can be defined in terms of the <a href="Inverse_image" class="mw-redirect" title="Inverse image">inverse images</a> of open sets, this means that the initial topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is given by taking all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}^{-1}(U),}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}^{-1}(U),}</annotation>

</semantics>

</math><img src="./7e85d8438f6130ab88c2907e880fdf94d74f422d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.892ex; height:3.343ex;" alt="{\displaystyle f_{i}^{-1}(U),}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> ranges over all open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{i},}</annotation>

</semantics>

</math><img src="./cf7f69fde10e4681cd2c1bd6da43865bb6a84321.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle Y_{i},}" loading="lazy"> as a subbasis.

Two important special cases of the initial topology are the <a href="Product_topology" title="Product topology">product topology</a>, where the family of functions is the set of projections from the product to each factor, and the <a href="Subspace_topology" title="Subspace topology">subspace topology</a>, where the family consists of just one function, the <a href="Inclusion_map" title="Inclusion map">inclusion map</a>.

The <a href="Compact-open_topology" title="Compact-open topology">compact-open topology</a> on the space of continuous functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> has for a subbase the set of functions

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(K,U)=\{f:X\to Y\mid f(K)\subseteq U\}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subseteq X}</annotation>

</semantics>

</math><img src="./4ba18595612961c1a8de8fbb2464e9bcfd4554da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.144ex; height:2.343ex;" alt="{\displaystyle K\subseteq X}" loading="lazy"> is <a href="Compact_space" title="Compact space">compact</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy">

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> is a <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> topological space with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> containing two or more elements (for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} }</annotation>

</semantics>

</math><img src="./1b2e2b6427cd2b517be352b378a1830c1540e3a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.757ex; height:2.176ex;" alt="{\displaystyle X=\mathbb {R} }" loading="lazy"> with the <a href="Euclidean_topology" title="Euclidean topology">Euclidean topology</a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\in \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\in \tau }</annotation>

</semantics>

</math><img src="./5a787a448f055b68339d47e1c35e7328c3e5c439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.816ex; height:2.176ex;" alt="{\displaystyle Y\in \tau }" loading="lazy"> be any non-empty open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> (for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> could be a non-empty bounded open interval in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">) and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> denote the <a href="Subspace_topology" title="Subspace topology">subspace topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> inherits from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> (so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \subseteq \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu \subseteq \tau }</annotation>

</semantics>

</math><img src="./05c20cb8bf0dbf36e16fe2583cb9c8bdec9212e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.533ex; height:2.176ex;" alt="{\displaystyle \nu \subseteq \tau }" loading="lazy">). Then the topology generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is equal to the union <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X\}\cup \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{X\}\cup \nu }</annotation>

</semantics>

</math><img src="./cf8a5529f71ee7461da78f68ff5494425c99af99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.12ex; height:2.843ex;" alt="{\displaystyle \{X\}\cup \nu }" loading="lazy"> (see the footnote for an explanation),

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X\}\cup \nu \subseteq \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{X\}\cup \nu \subseteq \tau }</annotation>

</semantics>

</math><img src="./ae1efdd0ad88cf55f1c6bfed06c63e6e4662a6ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.42ex; height:2.843ex;" alt="{\displaystyle \{X\}\cup \nu \subseteq \tau }" loading="lazy"> (since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> is Hausdorff, equality will hold if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=X}</annotation>

</semantics>

</math><img src="./213f9e0bf2b88042d7d90cf9e765ae8201743d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle Y=X}" loading="lazy">). Note that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a <a href="Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X\}\cup \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{X\}\cup \nu }</annotation>

</semantics>

</math><img src="./cf8a5529f71ee7461da78f68ff5494425c99af99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.12ex; height:2.843ex;" alt="{\displaystyle \{X\}\cup \nu }" loading="lazy"> is the smallest topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> yet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> does not cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (that is, the union <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{V\in \nu }V=Y}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigcup _{V\in \nu }V=Y}</annotation>

</semantics>

</math><img src="./1aefdf00630c9625c97dbda446dc1f47e950135b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.277ex; height:5.676ex;" alt="{\displaystyle \bigcup _{V\in \nu }V=Y}" loading="lazy"> is a proper subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">).

<div class="mw-heading mw-heading2"><h2 id="Results_using_subbases">Results using subbases</h2></div>

One nice fact about subbases is that <a href="Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">continuity</a> of a function need only be checked on a subbase of the range. That is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is a map between topological spaces and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> is a subbase for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

</semantics>

</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous <a href="If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(B)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(B)}</annotation>

</semantics>

</math><img src="./572ddad8cd0a0758fb98e1c94c432dc2f7a06636.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.227ex; height:3.176ex;" alt="{\displaystyle f^{-1}(B)}" loading="lazy"> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {B}}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {B}}.}</annotation>

</semantics>

</math><img src="./397bcc630152005bc86edb3b29338d6c7883b8a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.795ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {B}}.}" loading="lazy">

A <a href="Net_(mathematics)" title="Net (mathematics)">net</a> (or <a href="Sequence" title="Sequence">sequence</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}">

<msub>

</mrow>

</msub>

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}</annotation>

</semantics>

</math><img src="./5f06cbd9ae8f0977830b0571545f637115665c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.146ex; height:3.009ex;" alt="{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}" loading="lazy"> converges to a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> if and only if every subbasic neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> contains all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> for sufficiently large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\in I.}</annotation>

</semantics>

</math><img src="./d3627df37846ed8181157cbb3195735ec0988baa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.462ex; height:2.176ex;" alt="{\displaystyle i\in I.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Alexander_subbase_theorem">Alexander subbase theorem</h3></div>

The Alexander Subbase Theorem is a significant result concerning subbases that is due to <a href="James_Waddell_Alexander_II" title="James Waddell Alexander II">James Waddell Alexander II</a>.<a href="#cite_note-Muger2020-5">[3]</a> The corresponding result for basic (rather than subbasic) open covers is much easier to prove.

<dl><dd>Alexander subbase theorem:<a href="#cite_note-Muger2020-5">[3]</a><a href="#cite_note-FOOTNOTERudin1991392_Appendix_A2-1">[1]</a> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> be a topological space, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> be a subbase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

</semantics>

</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"> If every cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> by elements from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> has a finite subcover, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Compact_space" title="Compact space">compact</a>.</dd></dl>

The converse to this theorem also holds (because every cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> by elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> is an open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">)

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation>

</semantics>

</math><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"> be a topological space, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> be a subbase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

</semantics>

</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is compact, then every cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> by elements from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> has a finite subcover.</dd></dl>

/* start https://en.wikipedia.org/ */

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</style>

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:left;"><div style="font-size:115%;">Proof</div>

</th></tr>

<tr>

Suppose for the sake of contradiction that the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is not compact (so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is an infinite set), yet every subbasic cover from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> has a finite subcover.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation>

</semantics>

</math><img src="./9f9d5874c5d7f68eba1cec9da9ccbe53903303bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" loading="lazy"> denote the set of all open covers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that do not have any finite subcover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Partially order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation>

</semantics>

</math><img src="./9f9d5874c5d7f68eba1cec9da9ccbe53903303bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" loading="lazy"> by subset inclusion and use <a href="Zorn's_Lemma" class="mw-redirect" title="Zorn's Lemma">Zorn's Lemma</a> to find an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\in \mathbb {S} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\in \mathbb {S} }</annotation>

</semantics>

</math><img src="./ad267f2d2b199982cdaec5cf2dfc6e68bb184aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.372ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\in \mathbb {S} }" loading="lazy"> that is a maximal element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} .}</annotation>

</semantics>

</math><img src="./522b002a7daa6a8961cf43aa78e471df51ae3e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} .}" loading="lazy">

Observe that:

<ol><li>Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}</annotation>

</semantics>

</math><img src="./306385c890320899b17713f20e0bebe27db929b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.019ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}" loading="lazy"> by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ,}</annotation>

</semantics>

</math><img src="./31a6be518985e8ce458ddc7f49ea2d589da65f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle \mathbb {S} ,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> is an open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and there does not exist any finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> that covers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (so in particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> is infinite).</li>

<li>The maximality of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation>

</semantics>

</math><img src="./9f9d5874c5d7f68eba1cec9da9ccbe53903303bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" loading="lazy"> implies that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is an open set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\not \in {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\not \in {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./f938cf744f6e0d19cdb4ddf78abf2e217a172fca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.867ex; height:2.676ex;" alt="{\displaystyle V\not \in {\mathcal {C}}}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cup \{V\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cup \{V\}}</annotation>

</semantics>

</math><img src="./4ba470478e21605b69fb45c28d52309ce80df25f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.934ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}\cup \{V\}}" loading="lazy"> has a finite subcover, which must necessarily be of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V\}\cup {\mathcal {C}}_{V}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{V\}\cup {\mathcal {C}}_{V}}</annotation>

</semantics>

</math><img src="./da1faef8f53c4e3426d4d1df2ddfbb6573ca4789.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.416ex; height:2.843ex;" alt="{\displaystyle \{V\}\cup {\mathcal {C}}_{V}}" loading="lazy"> for some finite subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{V}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{V}}</annotation>

</semantics>

</math><img src="./ef445186654521d838f1ba03cadc054cd1dcbe45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.721ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{V}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> (this finite subset depends on the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">).</li></ol>

We will begin by showing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> is not a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> was a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> which in particular implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> is a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> by elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}.}</annotation>

</semantics>

</math><img src="./a109882bd79c5852d9c48a7e96f80f8fd7e0dcd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.139ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}.}" loading="lazy">

The theorem's hypothesis on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> implies that there exists a finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> that covers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> which would simultaneously also be a finite subcover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> by elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> (since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}\subseteq {\mathcal {C}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}\subseteq {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./8b6cdefcf889cceaf60dda76795501345c652320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.652ex; height:2.343ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}\subseteq {\mathcal {C}}}" loading="lazy">).

But this contradicts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}</annotation>

</semantics>

</math><img src="./306385c890320899b17713f20e0bebe27db929b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.019ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}\in \mathbb {S} ,}" loading="lazy"> which proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> does not cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> does not cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> that is not covered by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is not contained in any element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy">).

But since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> does cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> there also exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\in {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\in {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./be5c211124b5a799bd495707f9d5af47a91dda75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.862ex; height:2.176ex;" alt="{\displaystyle U\in {\mathcal {C}}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in U.}</annotation>

</semantics>

</math><img src="./2494bed786064aac20efd5ea0fcaa3f787dd6f14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.6ex; height:2.176ex;" alt="{\displaystyle x\in U.}" loading="lazy">

It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\neq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\neq X}</annotation>

</semantics>

</math><img src="./d39641c3ddcda37ae4d1c50c7b484fbb369f6544.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.861ex; height:2.676ex;" alt="{\displaystyle U\neq X}" loading="lazy">, because otherwise it would imply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> has a finite subcover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, namely the subcover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U\}=\{X\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{U\}=\{X\},}</annotation>

</semantics>

</math><img src="./404e3f5d3518c3047ba70200e5c68de40e8f6bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.158ex; height:2.843ex;" alt="{\displaystyle \{U\}=\{X\},}" loading="lazy"> contradicting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\in \mathbb {S} .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\in \mathbb {S} .}</annotation>

</semantics>

</math><img src="./baa79975bdcd2239389198dedbff416028c5a2a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.019ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\in \mathbb {S} .}" loading="lazy"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\neq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\neq X,}</annotation>

</semantics>

</math><img src="./380a60c0cf0b248f746083649ffd83441a538ab3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.508ex; height:2.676ex;" alt="{\displaystyle U\neq X,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> is a subbasis generating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">'s topology (together with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">), from the definition of the topology generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}},}</annotation>

</semantics>

</math><img src="./85f4ceaf9f455a0c9996bcf3467ac31815d8d8a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.139ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}},}" loading="lazy"> there must exist a finite collection of subbasic open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {S}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./334d51798b540c506bd8ee95bcac6bb2eec10676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.634ex; height:2.509ex;" alt="{\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {S}}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

</semantics>

</math><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy"> such that

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in S_{1}\cap \cdots \cap S_{n}\subseteq U.}</annotation>

</semantics>

</math>

We will now show by contradiction that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}\not \in {\mathcal {C}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{i}\not \in {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./0a074df8d8d652f24982476b6d62f8ece3b99aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.676ex;" alt="{\displaystyle S_{i}\not \in {\mathcal {C}}}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,n.}</annotation>

</semantics>

</math><img src="./df1fdaf4311ca9e5826a22477bcd28ce4b042fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> was such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}\in {\mathcal {C}},}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{i}\in {\mathcal {C}},}</annotation>

</semantics>

</math><img src="./15e928f44ff024b54a123b3429239108b899e337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.951ex; height:2.509ex;" alt="{\displaystyle S_{i}\in {\mathcal {C}},}" loading="lazy"> then also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}\in {\mathcal {C}}\cap {\mathcal {S}}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{i}\in {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./fb91cb8e5b0f0e46c3c83a0dee3b7e6ed4dd03e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.379ex; height:2.509ex;" alt="{\displaystyle S_{i}\in {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy"> so the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in S_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in S_{i}}</annotation>

</semantics>

</math><img src="./477d275b1a2a32cf1305643f34ebe6e3a155d9d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.395ex; height:2.509ex;" alt="{\displaystyle x\in S_{i}}" loading="lazy"> would then imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is covered by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}},}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}},}</annotation>

</semantics>

</math><img src="./4006154f11f0b437b2798393c2e7de1995a4ec20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.961ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}},}" loading="lazy"> which contradicts how <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> was chosen (recall that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> was chosen specifically so that it was not covered by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./43631fb2e1c4a1bfb45512ef53ad556888d94ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.314ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}" loading="lazy">).

As mentioned earlier, the maximality of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation>

</semantics>

</math><img src="./9f9d5874c5d7f68eba1cec9da9ccbe53903303bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" loading="lazy"> implies that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,n,}</annotation>

</semantics>

</math><img src="./959435349dbcf868b5468d0ae851972fc8833d75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n,}" loading="lazy"> there exists a finite subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{S_{i}}}">

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{S_{i}}}</annotation>

</semantics>

</math><img src="./ef08b3de4da9d79e8b41728d299ada226d79c93f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.09ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}_{S_{i}}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> such that<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{S_{i}}}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{S_{i}}}</annotation>

</semantics>

</math><img src="./8b8f3738a5d440027d1a91c408c44f8a5edb66b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.222ex; height:3.009ex;" alt="{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{S_{i}}}" loading="lazy"> forms a finite cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Define

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{F}:={\mathcal {C}}_{S_{1}}\cup \cdots \cup {\mathcal {C}}_{S_{n}},}</annotation>

</semantics>

</math>

which is a finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}.}</annotation>

</semantics>

</math><img src="./7dcc2530d6aa68953426cb5566491b4f00697872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.886ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}.}" loading="lazy">

Observe that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,n,}</annotation>

</semantics>

</math><img src="./959435349dbcf868b5468d0ae851972fc8833d75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./95ea45756779a21c55ac1b55e98806afe57157cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.82ex; height:2.843ex;" alt="{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}" loading="lazy"> is a finite cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> so let us replace every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{S_{i}}}">

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{S_{i}}}</annotation>

</semantics>

</math><img src="./ef08b3de4da9d79e8b41728d299ada226d79c93f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.09ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}_{S_{i}}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{F}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{F}.}</annotation>

</semantics>

</math><img src="./835efbe76a3efb96e9ee7f5cca2c05aeb5027c4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.335ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{F}.}" loading="lazy">

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup {\mathcal {C}}_{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./7c048b912b8134474bb16d5aa9e11332522fff88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.239ex; height:2.509ex;" alt="{\displaystyle \cup {\mathcal {C}}_{F}}" loading="lazy"> denote the union of all sets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./0f02ac78e36dbcb4b3b1a228c6f82e5795ea6129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.688ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{F}}" loading="lazy"> (which is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> denote the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup {\mathcal {C}}_{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./7c048b912b8134474bb16d5aa9e11332522fff88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.239ex; height:2.509ex;" alt="{\displaystyle \cup {\mathcal {C}}_{F}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Observe that for any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation>

</semantics>

</math><img src="./e8a030c36801bc01950928058517e7a7da7eaa08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A\}\cup {\mathcal {C}}_{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{A\}\cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./26d4ec158446d973ccd02ac2f6b628496babd764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.339ex; height:2.843ex;" alt="{\displaystyle \{A\}\cup {\mathcal {C}}_{F}}" loading="lazy"> covers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\subseteq A.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subseteq A.}</annotation>

</semantics>

</math><img src="./30ee0e305b44441417af3ff8ce923b45d9ce4c6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.169ex; height:2.343ex;" alt="{\displaystyle Z\subseteq A.}" loading="lazy">

In particular, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,n,}</annotation>

</semantics>

</math><img src="./959435349dbcf868b5468d0ae851972fc8833d75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.283ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n,}" loading="lazy"> the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./95ea45756779a21c55ac1b55e98806afe57157cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.82ex; height:2.843ex;" alt="{\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}" loading="lazy"> covers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\subseteq S_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subseteq S_{i}.}</annotation>

</semantics>

</math><img src="./e6f37ff62f8cb45ab523d59dc3b9d13bfb440450.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.65ex; height:2.509ex;" alt="{\displaystyle Z\subseteq S_{i}.}" loading="lazy">

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> was arbitrary, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\subseteq S_{1}\cap \cdots \cap S_{n}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subseteq S_{1}\cap \cdots \cap S_{n}.}</annotation>

</semantics>

</math><img src="./93b6f2ed5f6bf0aa0677004bd1f86efaf38d8692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.437ex; height:2.509ex;" alt="{\displaystyle Z\subseteq S_{1}\cap \cdots \cap S_{n}.}" loading="lazy">

Recalling that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}\cap \cdots \cap S_{n}\subseteq U,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}\cap \cdots \cap S_{n}\subseteq U,}</annotation>

</semantics>

</math><img src="./89c429a32f6907d676711822a0b67ca7f731689a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.539ex; height:2.509ex;" alt="{\displaystyle S_{1}\cap \cdots \cap S_{n}\subseteq U,}" loading="lazy"> we thus have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\subseteq U,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subseteq U,}</annotation>

</semantics>

</math><img src="./6d92e763826a37881be10b930d9a4606dfae24b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.208ex; height:2.509ex;" alt="{\displaystyle Z\subseteq U,}" loading="lazy"> which is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./2bdb19da512639db0ac8cca5d134027ef23cd93c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.378ex; height:2.843ex;" alt="{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}" loading="lazy"> being a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}</annotation>

</semantics>

</math><img src="./2bdb19da512639db0ac8cca5d134027ef23cd93c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.378ex; height:2.843ex;" alt="{\displaystyle \{U\}\cup {\mathcal {C}}_{F}}" loading="lazy"> is a finite cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U\}\cup {\mathcal {C}}_{F}\subseteq {\mathcal {C}}.}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{U\}\cup {\mathcal {C}}_{F}\subseteq {\mathcal {C}}.}</annotation>

</semantics>

</math><img src="./e75763a577668f9814a5432356bcf5e7d1dd08f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.363ex; height:2.843ex;" alt="{\displaystyle \{U\}\cup {\mathcal {C}}_{F}\subseteq {\mathcal {C}}.}" loading="lazy">

Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> has a finite subcover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> which contradicts the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\in \mathbb {S} .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\in \mathbb {S} .}</annotation>

</semantics>

</math><img src="./baa79975bdcd2239389198dedbff416028c5a2a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.019ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\in \mathbb {S} .}" loading="lazy">

Therefore, the original assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is not compact must be wrong, which proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is compact. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</td></tr></tbody></table></div>

Although this proof makes use of <a href="Zorn's_Lemma" class="mw-redirect" title="Zorn's Lemma">Zorn's Lemma</a>, the proof does not need the full strength of choice.

Instead, it relies on the intermediate <a href="Ultrafilter_principle" class="mw-redirect" title="Ultrafilter principle">Ultrafilter principle</a>.<a href="#cite_note-Muger2020-5">[3]</a>

Using this theorem with the subbase for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> above, one can give a very easy proof that bounded closed intervals in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> are compact.

More generally, <a href="Tychonoff's_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a>, which states that the product of non-empty compact spaces is compact, has a short proof if the Alexander Subbase Theorem is used.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:left;"><div style="font-size:115%;">Proof</div>

</th></tr>

<tr>

The product topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i}X_{i}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{i}X_{i}}</annotation>

</semantics>

</math><img src="./25951f7a02818dda8ba0c0a0224df04506bc6694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.08ex; height:5.509ex;" alt="{\displaystyle \prod _{i}X_{i}}" loading="lazy"> has, by definition, a subbase consisting of cylinder sets that are the inverse projections of an open set in one factor.

Given a subbasic family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> of the product that does not have a finite subcover, we can partition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\cup _{i}C_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C=\cup _{i}C_{i}}</annotation>

</semantics>

</math><img src="./911ae65f767c3655fd4dd7381d76d96dfcb6a093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.676ex; height:2.509ex;" alt="{\displaystyle C=\cup _{i}C_{i}}" loading="lazy"> into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space.

By assumption, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}\neq \varnothing }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}\neq \varnothing }</annotation>

</semantics>

</math><img src="./1f6f76445ea3da10b43a7d71e484502a1f68252b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.368ex; height:2.676ex;" alt="{\displaystyle C_{i}\neq \varnothing }" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation>

</semantics>

</math><img src="./cc49dc02c0ec8c86b67e7d10518ac791eda0bf22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}" loading="lazy"> does not have a finite subcover.

Being cylinder sets, this means their projections onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> have no finite subcover, and since each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> is compact, we can find a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in X_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}\in X_{i}}</annotation>

</semantics>

</math><img src="./c2685c5ed90347879276ed9996e9c759faec52e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.694ex; height:2.509ex;" alt="{\displaystyle x_{i}\in X_{i}}" loading="lazy"> that is not covered by the projections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation>

</semantics>

</math><img src="./cc49dc02c0ec8c86b67e7d10518ac791eda0bf22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}" loading="lazy"> onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}.}</annotation>

</semantics>

</math><img src="./6d8cf06c3e80129deea779a3d738464b1990906c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle X_{i}.}" loading="lazy"> But then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{i}\right)_{i}\in \prod _{i}X_{i}}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{i}\right)_{i}\in \prod _{i}X_{i}}</annotation>

</semantics>

</math><img src="./82330e2bf3dff0d5010e4c04cbfe30f48248c473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.659ex; height:5.509ex;" alt="{\displaystyle \left(x_{i}\right)_{i}\in \prod _{i}X_{i}}" loading="lazy"> is not covered by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C.}</annotation>

</semantics>

</math><img src="./067be67e68f60c53ce83241748d0d6249675c58d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

Note, that in the last step we implicitly used the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> (which is actually equivalent to <a href="Zorn's_lemma" title="Zorn's lemma">Zorn's lemma</a>) to ensure the existence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{i}\right)_{i}.}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{i}\right)_{i}.}</annotation>

</semantics>

</math><img src="./02398a20dd7f17dd5c286187559f40294d68228c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.385ex; height:3.009ex;" alt="{\displaystyle \left(x_{i}\right)_{i}.}" loading="lazy">

</td></tr></tbody></table></div>

<ul><li><a href="Base_(topology)" title="Base (topology)">Base (topology)</a> – Collection of open sets used to define a topology</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Rudin's definition is less general than ours, because it effectively requires that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> covers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (see "Alternative definition" subsection below). We drop this requirement here, and assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is any subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(X)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(X)}</annotation>

</semantics>

</math><img src="./f5ed5b6b7f1ad70cba0f7b3cf4603bf627321b5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.493ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(X)}" loading="lazy">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> is a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau ),}</annotation>

</semantics>

</math><img src="./2a40cd58d2e96a1b83e7713d38dd1576d14bb72e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle (X,\tau ),}" loading="lazy">, it is easy to verify that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X\}\cup \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{X\}\cup \nu }</annotation>

</semantics>

</math><img src="./cf8a5529f71ee7461da78f68ff5494425c99af99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.12ex; height:2.843ex;" alt="{\displaystyle \{X\}\cup \nu }" loading="lazy"> is a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> is closed under unions and finite intersections because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is. But since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\not \in \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\not \in \nu }</annotation>

</semantics>

</math><img src="./b866811c154b998e02eca674dc777d98d389ee85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.053ex; height:2.676ex;" alt="{\displaystyle X\not \in \nu }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> is not a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X\}\cup \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{X\}\cup \nu }</annotation>

</semantics>

</math><img src="./cf8a5529f71ee7461da78f68ff5494425c99af99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.12ex; height:2.843ex;" alt="{\displaystyle \{X\}\cup \nu }" loading="lazy"> is clearly the smallest topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">).

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>

<li id="cite_note-FOOTNOTERudin1991392_Appendix_A2-1">^ <a href="#cite_ref-FOOTNOTERudin1991392_Appendix_A2_1-0">a</a> <a href="#cite_ref-FOOTNOTERudin1991392_Appendix_A2_1-1">b</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 392 Appendix A2.

</li>

<li id="cite_note-FOOTNOTEMunkres200082-3"><a href="#cite_ref-FOOTNOTEMunkres200082_3-0">^</a> <a href="#CITEREFMunkres2000">Munkres 2000</a>, pp. 82.

</li>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFMuger2020" class="citation book cs1">Muger, Michael (2020). Topology for the Working Mathematician.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBourbaki1989" class="citation book cs1"><a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1989) [1966]. <a rel="nofollow" class="external text" href="https://doku.pub/documents/31425779-nicolas-bourbaki-general-topology-part-i1pdf-30j71z37920w">General Topology: Chapters 1–4</a> [<a href="Topologie_G%C3%A9n%C3%A9rale" class="mw-redirect" title="Topologie Générale">Topologie Générale</a>]. <a href="%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a>. Berlin New York: Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64241-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/18588129">18588129</a>.</cite></li>

<li><cite id="CITEREFDugundji1966" class="citation book cs1"><a href="James_Dugundji" title="James Dugundji">Dugundji, James</a> (1966). Topology. Boston: Allyn and Bacon. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-697-06889-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/395340485">395340485</a>.</cite></li>

<li><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). Topology (2nd ed.). <a href="Upper_Saddle_River%2C_NJ" class="mw-redirect" title="Upper Saddle River, NJ">Upper Saddle River, NJ</a>: <a href="Prentice_Hall%2C_Inc" class="mw-redirect" title="Prentice Hall, Inc">Prentice Hall, Inc</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-181629-9</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42683260">42683260</a>.</cite> (<a rel="nofollow" class="external text" href="https://archive.org/details/topology0002edmunk/page/n5/mode/2up">accessible to patrons with print disabilities</a>)</li>

<li><cite id="CITEREFRudin1991" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054236-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite></li>

<li><cite id="CITEREFWillard2004" class="citation book cs1">Willard, Stephen (2004) [1970]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-o8xJQ7Ag2cC">General Topology</a>. <a href="Mineola%2C_N.Y." class="mw-redirect" title="Mineola, N.Y.">Mineola, N.Y.</a>: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43479-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/115240">115240</a>.</cite></li></ul></div><div><div class="zim-footer">

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