SherbyNet - /page/wikibook/entry.mu

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<title>Uniform space</title>

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Uniform space

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In the <a href="Mathematical" class="mw-redirect" title="Mathematical">mathematical</a> field of <a href="Topology" title="Topology">topology</a>, a uniform space is a <a href="Topological_space" title="Topological space">set</a> with additional <a href="Mathematical_structure" title="Mathematical structure">structure</a> that is used to define <a href="Uniform_property" title="Uniform property">uniform properties</a>, such as <a href="Complete_space" class="mw-redirect" title="Complete space">completeness</a>, <a href="Uniform_continuity" title="Uniform continuity">uniform continuity</a> and <a href="Uniform_convergence" title="Uniform convergence">uniform convergence</a>. Uniform spaces generalize <a href="Metric_space" title="Metric space">metric spaces</a> and <a href="Topological_group" title="Topological group">topological groups</a>, but the concept is designed to formulate the weakest axioms needed for most proofs in <a href="Mathematical_analysis" title="Mathematical analysis">analysis</a>.

In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces. By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.

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

There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure.

<div class="mw-heading mw-heading3"><h3 id="Entourage_definition">Entourage definition</h3></div>

This definition adapts the presentation of a topological space in terms of <a href="Neighborhood_system" class="mw-redirect" title="Neighborhood system">neighborhood systems</a>. A nonempty collection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}">

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

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</math><img src="./0008a48abb9837a4cb0f495dd85d0ffda22ead92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}" loading="lazy"> is a <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>uniform structure (or a uniformity) if it satisfies the following axioms:

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

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</math><img src="./025d1cae24a1b5000987f91143629c3777ccc419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle U\in \Phi }" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq U,}">

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

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</math><img src="./637ca0e677ad2511c8d512621a1cdcfaeb21a5cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.464ex; height:2.509ex;" alt="{\displaystyle \Delta \subseteq U,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\{(x,x):x\in X\}}">

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</mrow>

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

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</math><img src="./162217290be6a735fa27c8acc024da86db79f0b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.949ex; height:2.843ex;" alt="{\displaystyle \Delta =\{(x,x):x\in X\}}" loading="lazy"> is the diagonal on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X.}">

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

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</math><img src="./6ef859a7a6d98b98c35000664b8b0d5b2e094b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.447ex; height:2.176ex;" alt="{\displaystyle X\times X.}" loading="lazy"></li>

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

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</math><img src="./025d1cae24a1b5000987f91143629c3777ccc419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle U\in \Phi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq V\subseteq X\times X}">

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

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</math><img src="./52f786a9345ce7831528db0bce518014538c97da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.567ex; height:2.343ex;" alt="{\displaystyle U\subseteq V\subseteq X\times X}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\in \Phi .}">

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

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</math><img src="./eb337e0d45abc11238599c722dd165f111d45977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.953ex; height:2.176ex;" alt="{\displaystyle V\in \Phi .}" loading="lazy"></li>

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</mrow>

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

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</math><img src="./025d1cae24a1b5000987f91143629c3777ccc419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle U\in \Phi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\in \Phi }">

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</mrow>

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

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</math><img src="./274371c51c6437e4bc5cc50d8a831575b39dbe2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.306ex; height:2.176ex;" alt="{\displaystyle V\in \Phi }" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap V\in \Phi .}">

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

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</math><img src="./3ecbda531640384b43609c4ca20a1ddc21d1ebf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.318ex; height:2.176ex;" alt="{\displaystyle U\cap V\in \Phi .}" loading="lazy"></li>

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</mrow>

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

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</math><img src="./025d1cae24a1b5000987f91143629c3777ccc419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle U\in \Phi }" loading="lazy"> then there is some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\in \Phi }">

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

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</math><img src="./274371c51c6437e4bc5cc50d8a831575b39dbe2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.306ex; height:2.176ex;" alt="{\displaystyle V\in \Phi }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\circ V\subseteq U}">

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

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</math><img src="./aa3233a2e32a433ad6e22f3587eaceadc6bfb9d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.65ex; height:2.343ex;" alt="{\displaystyle V\circ V\subseteq U}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\circ V}">

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

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</math><img src="./5e0a1c5d1f5b17082663ceee8a93f343a59e63b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.769ex; height:2.176ex;" alt="{\displaystyle V\circ V}" loading="lazy"> denotes the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</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"> with itself. The <a href="Composition_of_relations" title="Composition of relations">composite</a> of two subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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

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</mrow>

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

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

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

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</math><img src="./0008a48abb9837a4cb0f495dd85d0ffda22ead92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}" loading="lazy"> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\circ U=\{(x,z)~:~{\text{ there exists }}y\in X\,{\text{ such that }}\,(x,y)\in U\wedge (y,z)\in V\,\}.}">

<mtext> there exists </mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\circ U=\{(x,z)~:~{\text{ there exists }}y\in X\,{\text{ such that }}\,(x,y)\in U\wedge (y,z)\in V\,\}.}</annotation>

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</math></li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./025d1cae24a1b5000987f91143629c3777ccc419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle U\in \Phi }" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{-1}\in \Phi ,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U^{-1}\in \Phi ,}</annotation>

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</math><img src="./849f7fcf30e873ca8615da12e15b1de66f45aae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.34ex; height:3.009ex;" alt="{\displaystyle U^{-1}\in \Phi ,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{-1}=\{(y,x):(x,y)\in U\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./255fc5d9f13df9e2be636c741110362dd7142439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.815ex; height:3.176ex;" alt="{\displaystyle U^{-1}=\{(y,x):(x,y)\in U\}}" loading="lazy"> is the <a href="Converse_relation" title="Converse relation">inverse</a> of <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="./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>

The non-emptiness of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> taken together with (2) and (3) states that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is a <a href="Filter_(set_theory)" title="Filter (set theory)">filter</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6ef859a7a6d98b98c35000664b8b0d5b2e094b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.447ex; height:2.176ex;" alt="{\displaystyle X\times X.}" loading="lazy"> If the last property is omitted we call the space quasiuniform. An element <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"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is called a vicinity or entourage from the <a href="French_language" title="French language">French</a> word for surroundings.

One usually writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U[x]=\{y:(x,y)\in U\}=\operatorname {pr} _{2}(U\cap (\{x\}\times X)\,),}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U[x]=\{y:(x,y)\in U\}=\operatorname {pr} _{2}(U\cap (\{x\}\times X)\,),}</annotation>

</semantics>

</math><img src="./2cc8c4daeb651d21c0c79230f02c2f26a338feba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.723ex; height:2.843ex;" alt="{\displaystyle U[x]=\{y:(x,y)\in U\}=\operatorname {pr} _{2}(U\cap (\{x\}\times X)\,),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap (\{x\}\times X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4dbf3d174fe361598ccbf5e9eb9b3347e0a8b314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.649ex; height:2.843ex;" alt="{\displaystyle U\cap (\{x\}\times X)}" loading="lazy"> is the vertical cross section of <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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {pr} _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5c09c27559fff633dd39b11126638429dd38de13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.258ex; height:2.176ex;" alt="{\displaystyle \operatorname {pr} _{2}}" loading="lazy"> is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "<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="./d0abe2e7da593fb7b41d44e87a97fefdd8998b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}" loading="lazy">" diagonal; all the different <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="./6d52240d1f7a9ab0d4440551549d08445f99356f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle U[x]}" loading="lazy">'s form the vertical cross-sections. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)\in U}</annotation>

</semantics>

</math><img src="./7889b51b4276feffdcefc5c4fbb0ed85eaa567b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.952ex; height:2.843ex;" alt="{\displaystyle (x,y)\in U}" loading="lazy"> then one says 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"> 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="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> are <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">-close. Similarly, if all pairs of points in a subset <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="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> 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"> are <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">-close (that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./91e587030076802eec026dc75906339cf1f61b70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.327ex; height:2.176ex;" alt="{\displaystyle A\times A}" loading="lazy"> is contained in <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">), <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="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is called <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">-small. An entourage <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 symmetric if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)\in U}</annotation>

</semantics>

</math><img src="./7889b51b4276feffdcefc5c4fbb0ed85eaa567b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.952ex; height:2.843ex;" alt="{\displaystyle (x,y)\in U}" loading="lazy"> precisely when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,x)\in U.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a42108a78ea4a0a87ff2eb192c6a0f618c41dec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.598ex; height:2.843ex;" alt="{\displaystyle (y,x)\in U.}" loading="lazy"> The first axiom states that each point is <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">-close to itself for each entourage <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="./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"> The third axiom guarantees that being "both <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">-close and <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">-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage <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"> there is an entourage <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"> that is "not more than half as large". Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric 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="./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 <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="./83f72471aff7c6fbb27df0f971283a068efe091f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}" loading="lazy">

A base of entourages or fundamental system of entourages (or vicinities) of a uniformity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is any set <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"> of entourages of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> such that every entourage of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> contains a set belonging to <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"> Thus, by property 2 above, a fundamental systems of entourages <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 enough to specify the uniformity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> unambiguously: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is the set of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0008a48abb9837a4cb0f495dd85d0ffda22ead92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}" loading="lazy"> that contain a set of <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"> Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

Intuition about uniformities is provided by the example of <a href="Metric_space" title="Metric space">metric spaces</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cb4d7a16bca9e216c0221b43a1c3377aa5e358b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}" loading="lazy"> is a metric space, the sets

<msub>

</mrow>

</msub>

<mtext>where</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{a}=\{(x,y)\in X\times X:d(x,y)\leq a\}\quad {\text{where}}\quad a>0}</annotation>

</semantics>

</math>

form a fundamental system of entourages for the standard uniform structure 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"> 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="./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 <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="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9565fd4c69f836f3389927436673abb8ef229985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.689ex; height:2.509ex;" alt="{\displaystyle U_{a}}" loading="lazy">-close precisely when the distance between <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"> 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="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> is at most <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">

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is finer than another uniformity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.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 \Psi }" loading="lazy"> on the same set if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \supseteq \Psi ;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi \supseteq \Psi ;}</annotation>

</semantics>

</math><img src="./8599fd00ad1d2c0c3578a8a34b1ad34b99c334a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.232ex; height:2.509ex;" alt="{\displaystyle \Phi \supseteq \Psi ;}" loading="lazy"> in that case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.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 \Psi }" loading="lazy"> is said to be coarser than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi .}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./393f758b245feec1da90af1c2b4dfbbdab096d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \Phi .}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Pseudometrics_definition">Pseudometrics definition</h3></div>

Uniform spaces may be defined alternatively and equivalently using systems of <a href="Pseudometric_space" title="Pseudometric space">pseudometrics</a>, an approach that is particularly useful in <a href="Functional_analysis" title="Functional analysis">functional analysis</a> (with pseudometrics provided by <a href="Seminorm" title="Seminorm">seminorms</a>). More precisely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\times X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./228b01621f85b92004e028afb3dd9fd1757c3147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.308ex; height:2.509ex;" alt="{\displaystyle f:X\times X\to \mathbb {R} }" loading="lazy"> be a pseudometric on a set <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"> The inverse images <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{a}=f^{-1}([0,a])}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{a}=f^{-1}([0,a])}</annotation>

</semantics>

</math><img src="./e35532817972ff65d8d293addccc6c7d2455283a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.97ex; height:3.176ex;" alt="{\displaystyle U_{a}=f^{-1}([0,a])}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a>0}</annotation>

</semantics>

</math><img src="./1f34a80ea013edb56e340b19550430a8b6dfd7b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}" loading="lazy"> can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9565fd4c69f836f3389927436673abb8ef229985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.689ex; height:2.509ex;" alt="{\displaystyle U_{a}}" loading="lazy"> is the uniformity defined by the single pseudometric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"> Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces.

For a family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f_{i}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6682843675922a708cfb18c9661230473a88b930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.748ex; height:2.843ex;" alt="{\displaystyle \left(f_{i}\right)}" loading="lazy"> of pseudometrics 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="./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"> the uniform structure defined by the family is the least upper bound of the uniform structures defined by the individual pseudometrics <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="./efdb7feffe027e8a5affff98170471f67ac77aa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.586ex; height:2.509ex;" alt="{\displaystyle f_{i}.}" loading="lazy"> A fundamental system of entourages of this uniformity is provided by the set of finite intersections of entourages of the uniformities defined by the individual pseudometrics <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="./efdb7feffe027e8a5affff98170471f67ac77aa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.586ex; height:2.509ex;" alt="{\displaystyle f_{i}.}" loading="lazy"> If the family of pseudometrics is finite, it can be seen that the same uniform structure is defined by a single pseudometric, namely the <a href="Upper_envelope" class="mw-redirect" title="Upper envelope">upper envelope</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{}f_{i}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./69692085dde911267d8a84f3e8fa958175c2a63e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.827ex; height:4.176ex;" alt="{\displaystyle \sup _{}f_{i}}" loading="lazy"> of the family.

Less trivially, it can be shown that a uniform structure that admits a <a href="Countable" class="mw-redirect" title="Countable">countable</a> fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that any uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).

<div class="mw-heading mw-heading3"><h3 id="Uniform_cover_definition">Uniform cover definition</h3></div>

A uniform space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\Theta )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./edad0608c8bff686837247c7e14378e06c7d7b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\Theta )}" loading="lazy"> is a set <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"> equipped with a distinguished family of coverings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta ,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./903187858a4411b3814a69c019120c4a997463d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \Theta ,}" loading="lazy"> called "uniform covers", drawn from the set of <a href="Cover_(topology)" title="Cover (topology)">coverings</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"> that form a <a href="Filter_(mathematics)#General_definition:_Filter_on_a_partially_ordered_set" title="Filter (mathematics)">filter</a> when ordered by star refinement. One says that a cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> is a <a href="Star_refinement" title="Star refinement">star refinement</a> of cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./28f0145d00ea20b15431dee340a5b3ef5a8d3216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.655ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} ,}" loading="lazy"> written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} <^{*}\mathbf {Q} ,}">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} <^{*}\mathbf {Q} ,}</annotation>

</semantics>

</math><img src="./63944c250465106be68d0d3efba65774c2f90e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.634ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} <^{*}\mathbf {Q} ,}" loading="lazy"> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbf {P} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in \mathbf {P} ,}</annotation>

</semantics>

</math><img src="./8c435f6f81f89d954a2df586dc252765751f3e90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.057ex; height:2.509ex;" alt="{\displaystyle A\in \mathbf {P} ,}" loading="lazy"> there is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\in \mathbf {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./04b63984286b8b34695cc64f2e10807fce5eaf3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.631ex; height:2.509ex;" alt="{\displaystyle U\in \mathbf {Q} }" loading="lazy"> such that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B\neq \varnothing ,B\in \mathbf {P} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\cap B\neq \varnothing ,B\in \mathbf {P} ,}</annotation>

</semantics>

</math><img src="./78622b7f373cc4766f2a1292dd5396825f56302c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.108ex; height:2.676ex;" alt="{\displaystyle A\cap B\neq \varnothing ,B\in \mathbf {P} ,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq U.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b855f72666f6cf57a16b24d20c0d86446bdf1a8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.292ex; height:2.343ex;" alt="{\displaystyle B\subseteq U.}" loading="lazy"> Axiomatically, the condition of being a filter reduces to:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./79b34cea0172922a2b422eb792be091748edacea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.305ex; height:2.843ex;" alt="{\displaystyle \{X\}}" loading="lazy"> is a uniform cover (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X\}\in \Theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5e88c5f5024a13296fe760f2ba12018cfbcab393.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.954ex; height:2.843ex;" alt="{\displaystyle \{X\}\in \Theta }" loading="lazy">).</li>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} <^{*}\mathbf {Q} }</annotation>

</semantics>

</math><img src="./701b4d3368a7a6b7678b2825d0555af942f7f1bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.987ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} <^{*}\mathbf {Q} }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> a uniform cover and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132d0144479d6f47c30ad82a65d458966ccbe928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }" loading="lazy"> 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"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132d0144479d6f47c30ad82a65d458966ccbe928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }" loading="lazy"> is also a uniform cover.</li>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132d0144479d6f47c30ad82a65d458966ccbe928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }" loading="lazy"> are uniform covers then there is a uniform cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"> that star-refines both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132d0144479d6f47c30ad82a65d458966ccbe928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }" loading="lazy"></li></ol>

Given 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"> and a uniform cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59166319022da720747d775a5a442ccfa5ce217b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} ,}" loading="lazy"> one can consider the union of the members of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> that contain <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"> as a typical neighbourhood 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"> of "size" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59166319022da720747d775a5a442ccfa5ce217b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} ,}" loading="lazy"> and this intuitive measure applies uniformly over the space.

Given a uniform space in the entourage sense, define a cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> to be uniform if there is some entourage <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"> such that for each <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="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> there is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./21b670de721c35eb55c03d01cc377860348adbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\displaystyle A\in \mathbf {P} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U[x]\subseteq A.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b42f484dca7ce47cd6a977763a0d4f3cf3172998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.894ex; height:2.843ex;" alt="{\displaystyle U[x]\subseteq A.}" loading="lazy"> These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup \{A\times A:A\in \mathbf {P} \},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigcup \{A\times A:A\in \mathbf {P} \},}</annotation>

</semantics>

</math><img src="./f28df6f6808c629f6d62ceadb021c61867b7d9f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.228ex; height:3.843ex;" alt="{\displaystyle \bigcup \{A\times A:A\in \mathbf {P} \},}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"> ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other. <a href="#cite_note-isarmathlib_UniformSpace_ZF_2-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Topology_of_uniform_spaces">Topology of uniform spaces</h2></div>

Every uniform 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"> becomes a <a href="Topological_space" title="Topological space">topological space</a> by defining a nonempty subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\subseteq X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./04e2a448349ff1af6b692d8beb6d4d60b246c42d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle O\subseteq X}" loading="lazy"> to be open if and only if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in O}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8c221eb65d4c3d538e998233404ad50d9cd3e47a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.944ex; height:2.176ex;" alt="{\displaystyle x\in O}" loading="lazy"> there exists an entourage <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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[x]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bd4d3b31c894c1fd0b63162ae4b346783b390f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle V[x]}" loading="lazy"> is a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d1cdb120fda7ea217dfd4d6b9e297a863f45de87.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 O.}" loading="lazy"> In this topology, the neighbourhood filter of 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"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V[x]:V\in \Phi \}.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{V[x]:V\in \Phi \}.}</annotation>

</semantics>

</math><img src="./7cbfb5ad9a4ef6eae7305c806a407081e40698ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.625ex; height:2.843ex;" alt="{\displaystyle \{V[x]:V\in \Phi \}.}" loading="lazy"> This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[x]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bd4d3b31c894c1fd0b63162ae4b346783b390f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle V[x]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[y]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./02427391e6e67df71ffcfb4479bafbd5ed70a088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.236ex; height:2.843ex;" alt="{\displaystyle V[y]}" loading="lazy"> are considered to be of the "same size".

The topology defined by a uniform structure is said to be induced by the uniformity. A uniform structure on a topological space is compatible with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given 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="./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">

<div class="mw-heading mw-heading3"><h3 id="Uniformizable_spaces">Uniformizable spaces</h3></div>

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

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Uniformizable_space" title="Uniformizable space">Uniformizable space</a></div>

A topological space is called uniformizable if there is a uniform structure compatible with the topology.

Every uniformizable space is a <a href="Completely_regular_space" class="mw-redirect" title="Completely regular space">completely regular</a> topological space. Moreover, for a uniformizable 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"> the following are equivalent:

</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 <a href="Kolmogorov_space" title="Kolmogorov space">Kolmogorov space</a></li>

</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 <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a></li>

</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 <a href="Tychonoff_space" title="Tychonoff space">Tychonoff space</a></li>

<li>for any compatible uniform structure, the intersection of all entourages is the diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,x):x\in X\}.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fa98a6158ef3b23176af347c79c944cb04fb6043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.562ex; height:2.843ex;" alt="{\displaystyle \{(x,x):x\in X\}.}" loading="lazy"></li></ul>

Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.

The topology of a uniformizable space is always a <a href="Symmetric_topology" class="mw-redirect" title="Symmetric topology">symmetric topology</a>; that is, the space is an <a href="R0_space" class="mw-redirect" title="R0 space">R0-space</a>.

Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular 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"> can be defined as the coarsest uniformity that makes all continuous real-valued functions 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"> uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\times f)^{-1}(V),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a69026e96e7a53bfb3f3be2eed79cfde16a7b4b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.783ex; height:3.176ex;" alt="{\displaystyle (f\times f)^{-1}(V),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a continuous real-valued function 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 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 entourage of the uniform space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a067bb21dcf0642bdce48f05a55e218efab3b85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.65ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} .}" loading="lazy"> This uniformity defines a topology, which is clearly coarser than the original topology 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="./382fc9d9db960b232f5960d73b4a4762c7a047e0.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"> that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any <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"> and a neighbourhood <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"> 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="./feff4d40084c7351bf57b11ba2427f6331f5bdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}" loading="lazy"> there is a continuous real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=0}</annotation>

</semantics>

</math><img src="./cf85883d74b75fe35ca8d3f2b44802df078e4fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}" loading="lazy"> and equal to 1 in the complement 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="./1b2661a49b86bd1a5548e527bbfb068aa9f59978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}" loading="lazy">

In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff 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"> the set of all neighbourhoods of the diagonal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0008a48abb9837a4cb0f495dd85d0ffda22ead92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}" loading="lazy"> form the unique uniformity compatible with the topology.

A Hausdorff uniform space is <a href="Metrizable_space" title="Metrizable space">metrizable</a> if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed <a href="#Pseudometrics_definition">above</a>, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a <a href="Vector_space" title="Vector space">vector space</a> is Hausdorff and definable by a countable family of <a href="Seminorm" title="Seminorm">seminorms</a>, it is metrizable.

<div class="mw-heading mw-heading2"><h2 id="Uniform_continuity">Uniform continuity</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Uniform_continuity" title="Uniform continuity">Uniform continuity</a></div>

Similar to <a href="Continuous_function" title="Continuous function">continuous functions</a> between <a href="Topological_space" title="Topological space">topological spaces</a>, which preserve <a href="Topological_properties" class="mw-redirect" title="Topological properties">topological properties</a>, are the <a href="Uniformly_continuous_function" class="mw-redirect" title="Uniformly continuous function">uniformly continuous functions</a> between uniform spaces, which preserve uniform properties.

A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers. Explicitly, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

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

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</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"> between uniform spaces is called uniformly continuous if for every entourage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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

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

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</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"> there exists an entourage <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="./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"> in <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"> such that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{1},x_{2}\right)\in U}">

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<annotation encoding="application/x-tex">{\displaystyle \left(x_{1},x_{2}\right)\in U}</annotation>

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</math><img src="./06893ed308e2a228de8dfae5a249a5a4a7e5f347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.234ex; height:2.843ex;" alt="{\displaystyle \left(x_{1},x_{2}\right)\in U}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f\left(x_{1}\right),f\left(x_{2}\right)\right)\in V;}">

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<annotation encoding="application/x-tex">{\displaystyle \left(f\left(x_{1}\right),f\left(x_{2}\right)\right)\in V;}</annotation>

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</math><img src="./1d24046b25da93422a28ff95654defc264f5a77d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.223ex; height:2.843ex;" alt="{\displaystyle \left(f\left(x_{1}\right),f\left(x_{2}\right)\right)\in V;}" loading="lazy"> or in other words, whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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

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

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</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"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\times f)^{-1}(V)}">

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

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</math><img src="./14b8bf8390d8649811ada6044d3c5c6c0a3ffb7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.136ex; height:3.176ex;" alt="{\displaystyle (f\times f)^{-1}(V)}" loading="lazy"> is an entourage in <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">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\times f:X\times X\to Y\times Y}">

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

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</math><img src="./0fc1800d9864e708d16551d5eeb0fdf9cb3fd001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.136ex; height:2.509ex;" alt="{\displaystyle f\times f:X\times X\to Y\times Y}" loading="lazy"> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\times f)\left(x_{1},x_{2}\right)=\left(f\left(x_{1}\right),f\left(x_{2}\right)\right).}">

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<annotation encoding="application/x-tex">{\displaystyle (f\times f)\left(x_{1},x_{2}\right)=\left(f\left(x_{1}\right),f\left(x_{2}\right)\right).}</annotation>

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</math><img src="./a0d96a6ce88e85405ea30bdf7cfdf2ca6f710306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.286ex; height:2.843ex;" alt="{\displaystyle (f\times f)\left(x_{1},x_{2}\right)=\left(f\left(x_{1}\right),f\left(x_{2}\right)\right).}" loading="lazy">

All uniformly continuous functions are continuous with respect to the induced topologies.

Uniform spaces with uniform maps form a <a href="Category_(mathematics)" title="Category (mathematics)">category</a>. An <a href="Isomorphism" title="Isomorphism">isomorphism</a> between uniform spaces is called a <a href="Uniform_isomorphism" title="Uniform isomorphism">uniform isomorphism</a>; explicitly, it is a <a href="#uniformly_continuous">uniformly continuous</a> <a href="Bijection" title="Bijection">bijection</a> whose <a href="Inverse_function" title="Inverse function">inverse</a> is also uniformly continuous.

A uniform embedding is an injective uniformly continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\to Y}">

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

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</math><img src="./976e2f3fea37ed92af6613865d979dc22b272d31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.107ex; height:2.176ex;" alt="{\displaystyle i:X\to Y}" loading="lazy"> between uniform spaces whose inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{-1}:i(X)\to X}">

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<annotation encoding="application/x-tex">{\displaystyle i^{-1}:i(X)\to X}</annotation>

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</math><img src="./44ea6dd6db1a08637acd4d89201136594c1607d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.258ex; height:3.176ex;" alt="{\displaystyle i^{-1}:i(X)\to X}" loading="lazy"> is also uniformly continuous, where the image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(X)}">

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

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</math><img src="./26269fdc4e456980519b09ee45f2c990dc483c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle i(X)}" loading="lazy"> has the subspace uniformity inherited from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

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

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</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">

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

Generalizing the notion of <a href="Complete_metric_space" title="Complete metric space">complete metric space</a>, one can also define completeness for uniform spaces. Instead of working with <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a>, one works with <a href="Cauchy_filter" class="mw-redirect" title="Cauchy filter">Cauchy filters</a> (or <a href="Cauchy_net" class="mw-redirect" title="Cauchy net">Cauchy nets</a>).

A Cauchy filter (respectively, a Cauchy prefilter) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> on a uniform space <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"> is a <a href="Filter_(set_theory)" title="Filter (set theory)">filter</a> (respectively, a <a href="Prefilter" class="mw-redirect" title="Prefilter">prefilter</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> such that for every entourage <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="./59c21c569e498e0197798e3428b2bc6f25c0a457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U,}" loading="lazy"> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in F}">

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

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</math><img src="./c282e52300c61f5c1390d740735f48fe220d8318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle A\in F}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times A\subseteq U.}">

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

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</math><img src="./0a83cbaf91c397c1da3d93842af22a09732beacd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.854ex; height:2.343ex;" alt="{\displaystyle A\times A\subseteq U.}" loading="lazy"> In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter.

A minimal Cauchy filter is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique minimal Cauchy filter. The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.

Conversely, a uniform space is called complete if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.

Complete uniform spaces enjoy the following important property: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to Y}">

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

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</math><img src="./f6c5cc5035859bc6ae06d59f3dfebc17d6d716e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.346ex; height:2.509ex;" alt="{\displaystyle f:A\to Y}" loading="lazy"> is a uniformly continuous function from a <a href="Dense_set" title="Dense set">dense subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of a uniform space <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"> into a complete uniform space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

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

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</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}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> can be extended (uniquely) into a uniformly continuous function on all 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="./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 topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a <a href="Completely_uniformizable_space" title="Completely uniformizable space">completely uniformizable space</a>.

A completion of a uniform space <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"> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,C)}">

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</math><img src="./6f909755e3327bfe23b941265aec76d95f2a938e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.412ex; height:2.843ex;" alt="{\displaystyle (i,C)}" loading="lazy"> consisting of a complete uniform space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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</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"> and a <a href="#uniform_embedding">uniform embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\to C}">

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

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</math><img src="./6148d0ffbd04404a92163cd4217585900aa7692b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.1ex; height:2.176ex;" alt="{\displaystyle i:X\to C}" loading="lazy"> whose image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(X)}">

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</math><img src="./26269fdc4e456980519b09ee45f2c990dc483c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle i(X)}" loading="lazy"> is a <a href="Dense_set" title="Dense set">dense subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}">

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

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</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">

<div class="mw-heading mw-heading3"><h3 id="Hausdorff_completion_of_a_uniform_space">Hausdorff completion of a uniform space</h3></div>

As with metric spaces, every uniform space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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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"> has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</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 a uniformly continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\to Y}">

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

</semantics>

</math><img src="./976e2f3fea37ed92af6613865d979dc22b272d31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.107ex; height:2.176ex;" alt="{\displaystyle i:X\to Y}" 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 a Hausdorff uniform space then <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"> is a <a href="Topological_embedding" class="mw-redirect" title="Topological embedding">topological embedding</a>) with the following property:

<dl><dd>for any uniformly continuous mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> 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"> into a complete Hausdorff uniform space <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="./8f923ae5bf14eb7ac2432711f736967c063e26d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.327ex; height:2.509ex;" alt="{\displaystyle Z,}" loading="lazy"> there is a unique uniformly continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to Z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./44e7e6d2116baaa7aa88d1adbf796ade5997a237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.121ex; height:2.509ex;" alt="{\displaystyle g:Y\to Z}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=gi.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e3fdaf2e3fda138e3667adcfac02f6b2ae2be533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.942ex; height:2.509ex;" alt="{\displaystyle f=gi.}" loading="lazy"></dd></dl>

The Hausdorff completion <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 unique up to isomorphism. As a set, <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"> can be taken to consist of the minimal Cauchy filters 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="./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"> As the neighbourhood filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (x)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9b7567a5bd62c298019a578d336fef0fa8cfbbb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.04ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} (x)}" loading="lazy"> of each 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"> 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"> is a minimal Cauchy filter, the map <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"> can be defined by mapping <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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (x).}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8acf3b2556aa23124981aa01481ec2479125707e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.687ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} (x).}" loading="lazy"> The map <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"> thus defined is in general not injective; in fact, the graph of the equivalence relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(x)=i(x')}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fbce33a6f444e0cab186a9b9c31f476dbe838667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.666ex; height:3.009ex;" alt="{\displaystyle i(x)=i(x')}" loading="lazy"> is the intersection of all entourages 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"> and thus <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"> is injective precisely when <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 Hausdorff.

The uniform structure 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"> is defined as follows: for each symmetric entourage <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"> (that is, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)\in V}</annotation>

</semantics>

</math><img src="./94a765d76f66c745840fbff14fe8fea1d10d51cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.956ex; height:2.843ex;" alt="{\displaystyle (x,y)\in V}" loading="lazy"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,x)\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y,x)\in V}</annotation>

</semantics>

</math><img src="./03789329401615146d7ddab0a747a480785cd39a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.956ex; height:2.843ex;" alt="{\displaystyle (y,x)\in V}" loading="lazy">), let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(V)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./41c4cecddccf29a31848fb521b2408c8958b720f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.363ex; height:2.843ex;" alt="{\displaystyle C(V)}" loading="lazy"> be the set of all pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F,G)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./324c52461580e402f3e7f4be7b59442ffb8c5a52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.411ex; height:2.843ex;" alt="{\displaystyle (F,G)}" loading="lazy"> of minimal Cauchy filters which have in common at least one <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">-small set. The sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(V)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./41c4cecddccf29a31848fb521b2408c8958b720f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.363ex; height:2.843ex;" alt="{\displaystyle C(V)}" loading="lazy"> can be shown to form a fundamental system of entourages; <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 equipped with the uniform structure thus defined.

The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./26269fdc4e456980519b09ee45f2c990dc483c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle i(X)}" loading="lazy"> is then a dense 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"> 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 Hausdorff, then <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"> is an isomorphism onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(X),}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ce9b719536c7877adb6cee3326b03261f116714f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.239ex; height:2.843ex;" alt="{\displaystyle i(X),}" loading="lazy"> and thus <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"> can be identified with a dense subset of its completion. Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./26269fdc4e456980519b09ee45f2c990dc483c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle i(X)}" loading="lazy"> is always Hausdorff; it is called the Hausdorff uniform space associated 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="./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"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> denotes the equivalence relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(x)=i(x'),}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fc6969dc4d815502fa3a6b2388f5cf259f6caa43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.313ex; height:3.009ex;" alt="{\displaystyle i(x)=i(x'),}" loading="lazy"> then the quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/R}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fd34222b597f30c8f61770e00bdf191a02729e32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:2.843ex;" alt="{\displaystyle X/R}" loading="lazy"> is homeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(X).}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./347e4529e553a7d76a324df5c554d7ca3e5d81b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.239ex; height:2.843ex;" alt="{\displaystyle i(X).}" loading="lazy">

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

<ol><li>Every <a href="Metric_space" title="Metric space">metric space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,d)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d78e6f2ddf5baee227ee2a9f164726ba0c23c263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (M,d)}" loading="lazy"> can be considered as a uniform space. Indeed, since a metric is a fortiori a pseudometric, the <a class="mw-selflink-fragment" href="#Pseudometrics_definition">pseudometric definition</a> furnishes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad U_{a}\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\leq a\}.}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad U_{a}\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\leq a\}.}</annotation>

</semantics>

</math><img src="./4eadc1d16055cb1ef710e6e91ec5c5b870d8c284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.187ex; height:3.176ex;" alt="{\displaystyle \qquad U_{a}\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\leq a\}.}" loading="lazy"></blockquote>This uniform structure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> generates the usual metric space topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b865c33e30eb83000cd6387517c66dbbf3c3df9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.089ex; height:2.176ex;" alt="{\displaystyle M.}" loading="lazy"> However, different metric spaces can have the same uniform structure (trivial example is provided by a constant multiple of a metric). This uniform structure produces also equivalent definitions of <a href="Uniform_continuity" title="Uniform continuity">uniform continuity</a> and <a href="Complete_metric_space" title="Complete metric space">completeness for metric spaces</a>.</li>

<li>Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}(x,y)=|x-y|}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{1}(x,y)=|x-y|}</annotation>

</semantics>

</math><img src="./7d7058235d8035aac9ac525f22785b0ea374f7ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.309ex; height:2.843ex;" alt="{\displaystyle d_{1}(x,y)=|x-y|}" loading="lazy"> be the usual metric on <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 d_{2}(x,y)=\left|e^{x}-e^{y}\right|.}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{2}(x,y)=\left|e^{x}-e^{y}\right|.}</annotation>

</semantics>

</math><img src="./47c5b3062dafa931b653db5ac057f743a0dfc9d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.247ex; height:2.843ex;" alt="{\displaystyle d_{2}(x,y)=\left|e^{x}-e^{y}\right|.}" loading="lazy"> Then both metrics induce the usual topology on <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="./0522388d36b55de7babe4bbfc49475eaf590c2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" loading="lazy"> yet the uniform structures are distinct, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,y):|x-y|<1\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x,y):|x-y|<1\}}</annotation>

</semantics>

</math><img src="./3559a8b120a872a752465c7ef42e82d040159c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.471ex; height:2.843ex;" alt="{\displaystyle \{(x,y):|x-y|<1\}}" loading="lazy"> is an entourage in the uniform structure for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}(x,y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{1}(x,y)}</annotation>

</semantics>

</math><img src="./6d48a08e28f06f8f80fa7e492a455da6b55466c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.592ex; height:2.843ex;" alt="{\displaystyle d_{1}(x,y)}" loading="lazy"> but not for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}(x,y).}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{2}(x,y).}</annotation>

</semantics>

</math><img src="./a558af5959095dc176a9e8f0c28a8974b23223c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.238ex; height:2.843ex;" alt="{\displaystyle d_{2}(x,y).}" loading="lazy"> Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function.</li>

<li>Every <a href="Topological_group" title="Topological group">topological group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> (in particular, every <a href="Topological_vector_space" title="Topological vector space">topological vector space</a>) becomes a uniform space if we define a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq G\times G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./240c7e80ec492e092298adc22a087b52d0bad228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.38ex; height:2.343ex;" alt="{\displaystyle V\subseteq G\times G}" loading="lazy"> to be an entourage if and only if it contains the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,y):x\cdot y^{-1}\in U\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x,y):x\cdot y^{-1}\in U\}}</annotation>

</semantics>

</math><img src="./12589cad1116afb6f8f4f3c7918e81ed85c45db9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.716ex; height:3.176ex;" alt="{\displaystyle \{(x,y):x\cdot y^{-1}\in U\}}" loading="lazy"> for some <a href="Neighbourhood_(topology)" class="mw-redirect" title="Neighbourhood (topology)">neighborhood</a> <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"> of the <a href="Identity_element" title="Identity element">identity element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"> This uniform structure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is called the right uniformity on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> because for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./76db749e92a69b27e81b370b6ec3a2a199420ee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.544ex; height:2.509ex;" alt="{\displaystyle a\in G,}" loading="lazy"> the right multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to x\cdot a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d64aa4e67fbdb6f71f41238fa8a3c795aa06e52c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.182ex; height:1.843ex;" alt="{\displaystyle x\to x\cdot a}" loading="lazy"> is <a class="mw-selflink-fragment" href="#Uniform_continuity">uniformly continuous</a> with respect to this uniform structure. One may also define a left uniformity on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G;}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d57f0e2433ff1624498378f0166f86c14d5086c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G;}" loading="lazy"> the two need not coincide, but they both generate the given topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"></li>

<li>For every topological group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and its subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subseteq G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6dbe10924c5c9134471b96d6a8dfb9c8b1bf0f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.989ex; height:2.343ex;" alt="{\displaystyle H\subseteq G}" loading="lazy"> the set of left <a href="Coset" title="Coset">cosets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> is a uniform space with respect to the uniformity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> defined as follows. The sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {U}}=\{(s,t)\in G/H\times G/H:\ \ t\in U\cdot s\},}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {U}}=\{(s,t)\in G/H\times G/H:\ \ t\in U\cdot s\},}</annotation>

</semantics>

</math><img src="./5fc55ab7f388c4b2902e58127004ed0fe8d80925.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.744ex; height:3.176ex;" alt="{\displaystyle {\tilde {U}}=\{(s,t)\in G/H\times G/H:\ \ t\in U\cdot s\},}" 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"> runs over neighborhoods of the identity in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> form a fundamental system of entourages for the uniformity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi .}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./393f758b245feec1da90af1c2b4dfbbdab096d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \Phi .}" loading="lazy"> The corresponding induced topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> is equal to the <a href="Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a> defined by the natural map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\to G/H.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\to G/H.}</annotation>

</semantics>

</math><img src="./33217ff6d1db5059bbeb0617164cfe30169f2a95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.43ex; height:2.843ex;" alt="{\displaystyle g\to G/H.}" loading="lazy"></li>

<li>The trivial topology belongs to a uniform space in which the whole cartesian product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0008a48abb9837a4cb0f495dd85d0ffda22ead92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}" loading="lazy"> is the only <a href="Entourage_(topology)" class="mw-redirect" title="Entourage (topology)">entourage</a>.</li></ol>

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

Before <a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a> gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using <a href="Metric_spaces" class="mw-redirect" title="Metric spaces">metric spaces</a>. <a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a> provided the definition of uniform structure in terms of entourages in the book <a href="Topologie_G%C3%A9n%C3%A9rale" class="mw-redirect" title="Topologie Générale">Topologie Générale</a> and <a href="John_Tukey" title="John Tukey">John Tukey</a> gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.

<ul><li><a href="Coarse_structure" title="Coarse structure">Coarse structure</a> – Concept in geometry and topology</li>

<li><a href="Complete_metric_space" title="Complete metric space">Complete metric space</a> – Metric geometry</li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete topological vector space</a> – Structure in functional analysis</li>

<li><a href="Completely_uniformizable_space" title="Completely uniformizable space">Completely uniformizable space</a></li>

<li><a href="Filters_in_topology" title="Filters in topology">Filters in topology</a> – Use of filters to describe and characterize all basic topological notions and results</li>

<li><a href="Initial_uniform_structure" class="mw-redirect" title="Initial uniform structure">Initial uniform structure</a> – Coarsest topology making certain functions continuousPages displaying short descriptions of redirect targets</li>

<li><a href="Proximity_space" title="Proximity space">Proximity space</a> – Structure describing a notion of "nearness" between subsets</li>

<li><a href="Space_(mathematics)" title="Space (mathematics)">Space (mathematics)</a> – Mathematical set with some added structure</li>

<li><a href="Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Topology of uniform convergence</a></li>

<li><a href="Uniform_continuity" title="Uniform continuity">Uniform continuity</a> – Uniform restraint of the change in functions</li>

<li><a href="Uniform_isomorphism" title="Uniform isomorphism">Uniform isomorphism</a> – Uniformly continuous homeomorphism</li>

<li><a href="Uniform_property" title="Uniform property">Uniform property</a> – Object of study in the category of uniform topological spaces</li>

<li><a href="Uniformly_connected_space" title="Uniformly connected space">Uniformly connected space</a> – Type of uniform space</li></ul>

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://isarmathlib.org/UniformSpace_ZF_2.html">"IsarMathLib.org"</a>. Retrieved 2021-10-02.</cite>

</li>

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

<ul><li><a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>, <cite>General Topology</cite> (<cite>Topologie Générale</cite>), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-19374-X</bdi> (Ch. 1–4), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-19372-3</bdi> (Ch. 5–10): Chapter II is a comprehensive reference of uniform structures, Chapter IX § 1 covers pseudometrics, and Chapter III § 3 covers uniform structures on topological groups</li>

<li><a href="Ryszard_Engelking" title="Ryszard Engelking">Ryszard Engelking</a>, <cite>General Topology. Revised and completed edition</cite>, Berlin 1989.</li>

<li><a href="John_R._Isbell" title="John R. Isbell">John R. Isbell</a>, <cite>Uniform Spaces</cite> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-1512-1</bdi></li>

<li><a href="Ioan_James" title="Ioan James">I. M. James</a>, <cite>Introduction to Uniform Spaces</cite> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-38620-9</bdi></li>

<li><a href="Ioan_James" title="Ioan James">I. M. James</a>, <cite>Topological and Uniform Spaces</cite> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-96466-5</bdi></li>

<li><a href="John_Tukey" title="John Tukey">John Tukey</a>, <cite>Convergence and Uniformity in Topology</cite>; <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-09568-X</bdi></li>

<li><a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a>, <cite>Sur les espaces à structure uniforme et sur la topologie générale</cite>, Act. Sci. Ind. 551, Paris, 1937</li></ul>

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</style><div id="Topology1035" style="font-size:114%;margin:0 4em"><a href="Topology" title="Topology">Topology</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_topology" title="General topology">General (point-set)</a></li>

<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a></li>

<li><a href="Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li>

<li><a href="Continuum_(topology)" title="Continuum (topology)">Continuum</a></li>

<li><a href="Differential_topology" title="Differential topology">Differential</a></li>

<li><a href="Geometric_topology" title="Geometric topology">Geometric</a>

<ul><li><a href="Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li>

<li><a href="Homology_(mathematics)" title="Homology (mathematics)">Homology</a>

<ul><li><a href="Cohomology" title="Cohomology">cohomology</a></li></ul></li>

<li><a href="Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>

<li><a href="Digital_topology" title="Digital topology">Digital</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><a href="Klein_bottle" title="Klein bottle"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Open_set" title="Open set">Open set</a> / <a href="Closed_set" title="Closed set">Closed set</a></li>

<li><a href="Interior_(topology)" title="Interior (topology)">Interior</a></li>

<li><a href="Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li>

<li><a href="Topological_space" title="Topological space">Space</a>

<ul><li><a href="Compact_space" title="Compact space">compact</a></li>

<li><a href="Connected_space" title="Connected space">connected</a></li>

<li><a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a></li>

<li><a href="Metric_space" title="Metric space">metric</a></li>

</ul></li>

<li><a href="Homotopy" title="Homotopy">Homotopy</a>

<ul><li><a href="Homotopy_group" title="Homotopy group">homotopy group</a></li>

<li><a href="Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li>

<li><a href="Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li>

<li><a href="CW_complex" title="CW complex">CW complex</a></li>

<li><a href="Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li>

<li><a href="Second-countable_space" title="Second-countable space">Second-countable space</a></li>

<li><a href="Cobordism" title="Cobordism">Cobordism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Metrics and properties</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li>

<li><a href="Betti_number" title="Betti number">Betti number</a></li>

<li><a href="Winding_number" title="Winding number">Winding number</a></li>

<li><a href="Chern_class" title="Chern class">Chern number</a></li>

<li><a href="Orientability" title="Orientability">Orientability</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li>

<li><a href="Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li>

<li><a href="Tychonoff's_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li>

<li><a href="Urysohn's_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li> Category</li>

<li> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li>

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<li> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw external" title="wikiversity:Topology">Wikiversity</a></li>

<li> <a href="List_of_topology_topics" title="List of topology topics">Topics</a>

<ul><li><a href="List_of_general_topology_topics" title="List of general topology topics">general</a></li>

<li><a href="List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li>

<li><a href="List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li>

<li> <a href="List_of_important_publications_in_mathematics" class="mw-redirect" title="List of important publications in mathematics">Publications</a></li></ul>

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