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Hyperconnected space

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</style><div role="note" class="hatnote navigation-not-searchable">For the computer networking term, see <a href="Hyperconnectivity" title="Hyperconnectivity">Hyperconnectivity</a>. For hyper-connectivity in node-link graphs, see <a href="Connectivity_(graph_theory)#Super-_and_hyper-connectivity" title="Connectivity (graph theory)">Connectivity (graph theory) § Super- and hyper-connectivity</a>.</div>

In the mathematical field of <a href="Topology" title="Topology">topology</a>, a hyperconnected space<a href="#cite_note-1">[1]</a><a href="#cite_note-FOOTNOTEHartNagataVaughan20049-2">[2]</a> or irreducible space<a href="#cite_note-FOOTNOTEHartNagataVaughan20049-2">[2]</a> is a <a href="Topological_space" title="Topological space">topological space</a> X that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). The name irreducible space is preferred in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>.

For a topological space X the following conditions are equivalent:

<ul><li>No two nonempty <a href="Open_set" title="Open set">open sets</a> are <a href="Disjoint_sets" title="Disjoint sets">disjoint</a>.</li>

<li>X cannot be written as the union of two proper <a href="Closed_subset" class="mw-redirect" title="Closed subset">closed subsets</a>.</li>

<li>Every nonempty open set is <a href="Dense_(topology)" class="mw-redirect" title="Dense (topology)">dense</a> in X.</li>

<li>Every open set is connected.</li>

<li>The <a href="Interior_(topology)" title="Interior (topology)">interior</a> of every proper closed subset of X is empty.</li>

<li>Every subset is dense or <a href="Nowhere_dense_set" title="Nowhere dense set">nowhere dense</a> in X.</li>

<li>No two points can be separated by disjoint neighbourhoods.</li></ul>

A space which satisfies any one of these conditions is called hyperconnected or irreducible. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> property, some authors call such spaces anti-Hausdorff.<a href="#cite_note-3">[3]</a>

The <a href="Empty_set" title="Empty set">empty set</a> is <a href="Vacuous_truth" title="Vacuous truth">vacuously</a> a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets). However some authors,<a href="#cite_note-4">[4]</a> especially those interested in applications to <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, add an explicit condition that an irreducible space must be nonempty.

An irreducible set is a subset of a topological space for which the <a href="Subspace_topology" title="Subspace topology">subspace topology</a> is irreducible.

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

Two examples of hyperconnected spaces from <a href="Point_set_topology" class="mw-redirect" title="Point set topology">point set topology</a> are the <a href="Cofinite_topology" class="mw-redirect" title="Cofinite topology">cofinite topology</a> on any <a href="Infinite_set" title="Infinite set">infinite set</a> and the <a href="Order_topology#Left_and_right_order_topologies" title="Order topology">right order topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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

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</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">.

In algebraic geometry, taking the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a> whose <a href="Reduced_ring" title="Reduced ring">reduced ring</a> is an <a href="Integral_domain" title="Integral domain">integral domain</a> is an irreducible topological space—applying the <a href="Lattice_theorem" class="mw-redirect" title="Lattice theorem">lattice theorem</a> to the <a href="Nilradical_of_a_ring" title="Nilradical of a ring">nilradical</a>, which is within every prime, to show the spectrum of the quotient map is a <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a>, this reduces to the irreducibility of the spectrum of an integral domain. For example, the <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a><blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}\left({\frac {\mathbb {Z} [x,y,z]}{x^{4}+y^{3}+z^{2}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}\left({\frac {\mathbb {Z} [x,y,z]}{x^{4}+y^{3}+z^{2}}}\right)}</annotation>

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</math><img src="./236f9480a7afe3e5f53da933e654249765472018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.718ex; height:6.343ex;" alt="{\displaystyle {\text{Spec}}\left({\frac {\mathbb {Z} [x,y,z]}{x^{4}+y^{3}+z^{2}}}\right)}" loading="lazy"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(y^{2}z-x(x-z)(x-2z))}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(y^{2}z-x(x-z)(x-2z))}}\right)}</annotation>

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</math><img src="./a410f772733410809e9ea40644e901f3bd8743f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.593ex; height:6.509ex;" alt="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(y^{2}z-x(x-z)(x-2z))}}\right)}" loading="lazy"></blockquote>are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial factorization). A non-example is given by the <a href="Normal_crossing_divisor" class="mw-redirect" title="Normal crossing divisor">normal crossing divisor</a><blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xyz)}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xyz)}}\right)}</annotation>

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</math><img src="./78c91e4f31c77ba2b4b8417c11fcd2f40889e2d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.907ex; height:6.509ex;" alt="{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xyz)}}\right)}" loading="lazy"></blockquote>since the underlying space is the union of the affine planes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{x,y}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{x,y}^{2}}</annotation>

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</math><img src="./2e24e897bb78d0b2ef4e655e3fcec5dfa786c88a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.125ex; height:3.509ex;" alt="{\displaystyle \mathbb {A} _{x,y}^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{x,z}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{x,z}^{2}}</annotation>

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</math><img src="./78d33a0130f6a51e3b76f9f698905e20b16005bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.077ex; height:3.509ex;" alt="{\displaystyle \mathbb {A} _{x,z}^{2}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{y,z}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{y,z}^{2}}</annotation>

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</math><img src="./8a8a8a3550faf4a1b4a10231165dad570c801cc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.954ex; height:3.509ex;" alt="{\displaystyle \mathbb {A} _{y,z}^{2}}" loading="lazy">. Another non-example is given by the scheme<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z,w]}{(xy,f_{4})}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z,w]}{(xy,f_{4})}}\right)}</annotation>

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</math><img src="./44d6191ff07ed2c626040bdf7d360dcbb7c0ced5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.324ex; height:6.509ex;" alt="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z,w]}{(xy,f_{4})}}\right)}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{4}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{4}}</annotation>

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</math><img src="./294f14bcf91a6ff040c4c22b71720764d800fc26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{4}}" loading="lazy"> is an irreducible degree 4 <a href="Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomial</a>. This is the union of the two genus 3 curves (by the <a href="Genus%E2%80%93degree_formula" title="Genus–degree formula">genus–degree formula</a>)<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [y,z,w]}{(f_{4}(0,y,z,w))}}\right),{\text{ }}{\text{Proj}}\left({\frac {\mathbb {C} [x,z,w]}{(f_{4}(x,0,z,w))}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [y,z,w]}{(f_{4}(0,y,z,w))}}\right),{\text{ }}{\text{Proj}}\left({\frac {\mathbb {C} [x,z,w]}{(f_{4}(x,0,z,w))}}\right)}</annotation>

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</math><img src="./a4e7c1b431573f2736759c68316071d96de341dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.17ex; height:6.509ex;" alt="{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [y,z,w]}{(f_{4}(0,y,z,w))}}\right),{\text{ }}{\text{Proj}}\left({\frac {\mathbb {C} [x,z,w]}{(f_{4}(x,0,z,w))}}\right)}" loading="lazy"></blockquote>

<div class="mw-heading mw-heading2"><h2 id="Hyperconnectedness_vs._connectedness">Hyperconnectedness vs. connectedness</h2></div>

Every hyperconnected space is both <a href="Connected_space" title="Connected space">connected</a> and <a href="Locally_connected" class="mw-redirect" title="Locally connected">locally connected</a> (though not necessarily <a href="Path-connected" class="mw-redirect" title="Path-connected">path-connected</a> or <a href="Locally_path-connected" class="mw-redirect" title="Locally path-connected">locally path-connected</a>).

Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to the definition of connectedness, in which the open sets are disjoint.

For example, the space of real numbers with the standard topology is connected but not hyperconnected. This is because it cannot be written as a union of two disjoint open sets, but it can be written as a union of two (non-disjoint) closed sets.

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

<ul><li>The nonempty open subsets of a hyperconnected space are "large" in the sense that each one is dense in X and any pair of them intersects. Thus, a hyperconnected space cannot be <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> unless it contains only a single point.</li>

<li>Every hyperconnected space is both <a href="Connected_space" title="Connected space">connected</a> and <a href="Locally_connected" class="mw-redirect" title="Locally connected">locally connected</a> (though not necessarily <a href="Path-connected" class="mw-redirect" title="Path-connected">path-connected</a> or <a href="Locally_path-connected" class="mw-redirect" title="Locally path-connected">locally path-connected</a>).</li>

<li>Since the closure of every non-empty open set in a hyperconnected space is the whole space, which is an open set, every hyperconnected space is <a href="Extremally_disconnected_space" title="Extremally disconnected space">extremally disconnected</a>.</li>

<li>The <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> image of a hyperconnected space is hyperconnected.<a href="#cite_note-5">[5]</a> In particular, any continuous function from a hyperconnected space to a Hausdorff space must be constant. It follows that every hyperconnected space is <a href="Pseudocompact_space" title="Pseudocompact space">pseudocompact</a>.</li>

<li>Every open subspace of a hyperconnected space is hyperconnected.<a href="#cite_note-6">[6]</a></li></ul>

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

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</math><img src="./c01cf5893c47ae0bfe4df06f73175c8d35bd68fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.861ex; height:2.176ex;" alt="{\displaystyle U\subset X}" loading="lazy"> be an open subset. Any two disjoint open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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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"> would themselves be disjoint open subsets of <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">. So at least one of them must be empty.</dd></dl>

<ul><li>More generally, every dense subset of a hyperconnected space is hyperconnected.</li></ul>

<dl><dd>Proof: Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is a dense subset of <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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=S_{1}\cup S_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle S=S_{1}\cup S_{2}}</annotation>

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</math><img src="./6225d78a9900c6d9c42f69bb7f86a5e30cccab09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.139ex; height:2.509ex;" alt="{\displaystyle S=S_{1}\cup S_{2}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

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</math><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

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

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</math><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy"> closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\overline {S}}={\overline {S_{1}}}\cup {\overline {S_{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle X={\overline {S}}={\overline {S_{1}}}\cup {\overline {S_{2}}}}</annotation>

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</math><img src="./944522ff5c8ef39f1516d0d7ba7a4712e43ec1e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.61ex; height:3.343ex;" alt="{\displaystyle X={\overline {S}}={\overline {S_{1}}}\cup {\overline {S_{2}}}}" loading="lazy">. Since <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"> is hyperconnected, one of the two closures is the whole 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">, say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S_{1}}}=X}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {S_{1}}}=X}</annotation>

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</math><img src="./cd842788f016c9fcbd2466b9af56c110deeb17d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.673ex; height:3.343ex;" alt="{\displaystyle {\overline {S_{1}}}=X}" loading="lazy">. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

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

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</math><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"> is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, and since it is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, it must be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">.</dd></dl>

<ul><li>A closed subspace of a hyperconnected space need not be hyperconnected.</li></ul>

<dl><dd>Counterexample: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Bbbk ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8a9882a72c4026f424b7d7815195e93d1d1ca7cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.347ex; height:2.676ex;" alt="{\displaystyle \Bbbk ^{2}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Bbbk }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cdba77686112891f2431f01b94d4738be861f15a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \Bbbk }" loading="lazy"> an <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> (thus infinite) is hyperconnected<a href="#cite_note-7">[7]</a> in the <a href="Zariski_topology" title="Zariski topology">Zariski topology</a>, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=Z(XY)=Z(X)\cup Z(Y)\subset \Bbbk ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=Z(XY)=Z(X)\cup Z(Y)\subset \Bbbk ^{2}}</annotation>

</semantics>

</math><img src="./1072ea4af4c56efc4f81d2e882a5a44609fc045f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.988ex; height:3.176ex;" alt="{\displaystyle V=Z(XY)=Z(X)\cup Z(Y)\subset \Bbbk ^{2}}" loading="lazy"> is closed and not hyperconnected.</dd></dl>

<ul><li>The <a href="Closure_(topology)" title="Closure (topology)">closure</a> of any irreducible set is irreducible.<a href="#cite_note-8">[8]</a></li></ul>

<dl><dd>Proof: Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./44aba72977e43f863dd873b095d1dc0bd3f17608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is irreducible and write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} _{X}(S)=F\cup G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} _{X}(S)=F\cup G}</annotation>

</semantics>

</math><img src="./86582789fa56658c557e61b0fb67d3a222c05b60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.514ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} _{X}(S)=F\cup G}" loading="lazy"> for two closed subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\subseteq \operatorname {Cl} _{X}(S)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F,G\subseteq \operatorname {Cl} _{X}(S)}</annotation>

</semantics>

</math><img src="./474200c52df98c04350ef796b91e4f9b10ecc70b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.966ex; height:2.843ex;" alt="{\displaystyle F,G\subseteq \operatorname {Cl} _{X}(S)}" loading="lazy"> (and thus 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">). <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F':=F\cap S,\,G':=G\cap S}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F':=F\cap S,\,G':=G\cap S}</annotation>

</semantics>

</math><img src="./33d238e63b4217e8ee3c857615e064439d9e00ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.654ex; height:2.843ex;" alt="{\displaystyle F':=F\cap S,\,G':=G\cap S}" loading="lazy"> are closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=F'\cup G'}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./28fd270113cc88464f4daf1fdd43e93d66673d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.191ex; height:2.509ex;" alt="{\displaystyle S=F'\cup G'}" loading="lazy"> which implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./20c3eac60b17788eb1d8fcb19a894c1ac8e2a3c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.338ex; height:2.343ex;" alt="{\displaystyle S\subseteq F}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./661adc96925066de10003e726886d8238cea7302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.424ex; height:2.343ex;" alt="{\displaystyle S\subseteq G}" loading="lazy">, but then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} _{X}(S)=F}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} _{X}(S)=F}</annotation>

</semantics>

</math><img src="./e6794ed5526c56b78607ab818a92a5e725dfc3ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.105ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} _{X}(S)=F}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} _{X}(S)=G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} _{X}(S)=G}</annotation>

</semantics>

</math><img src="./aa88b984cce1bdf1cbb5631a9f3a5e9274ec1015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.191ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} _{X}(S)=G}" loading="lazy"> by definition of <a href="Closure_(topology)" title="Closure (topology)">closure</a>.</dd></dl>

<ul><li>A 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"> which can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=U_{1}\cup U_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=U_{1}\cup U_{2}}</annotation>

</semantics>

</math><img src="./a76396252955799c2d0bb9ff3dae761c63c57d68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.945ex; height:2.509ex;" alt="{\displaystyle X=U_{1}\cup U_{2}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1},U_{2}\subset X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{1},U_{2}\subset X}</annotation>

</semantics>

</math><img src="./438d11a8f461c1669f55cd586363ef37c99ba55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.396ex; height:2.509ex;" alt="{\displaystyle U_{1},U_{2}\subset X}" loading="lazy"> open and irreducible such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}\cap U_{2}\neq \emptyset }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{1}\cap U_{2}\neq \emptyset }</annotation>

</semantics>

</math><img src="./1c8db4f548fabfa8dbd832bd87b499819cf59d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.127ex; height:2.843ex;" alt="{\displaystyle U_{1}\cap U_{2}\neq \emptyset }" loading="lazy"> is irreducible.<a href="#cite_note-9">[9]</a></li></ul>

<dl><dd>Proof: Firstly, we notice that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is a non-empty open set 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"> then it intersects both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bc9e7f892894bc50c32ce1b9f9a68a15562146ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./590fa6a550fbe2866a28243a733d54245d218b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{2}}" loading="lazy">; indeed, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}:=U_{1}\cap V\neq \emptyset }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}:=U_{1}\cap V\neq \emptyset }</annotation>

</semantics>

</math><img src="./8a14c73b818be5b3131551fef670d658adf0b911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.427ex; height:2.843ex;" alt="{\displaystyle V_{1}:=U_{1}\cap V\neq \emptyset }" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./adfdbc929f16cb00bb43289c223651b41f7b9f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}" loading="lazy"> is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bc9e7f892894bc50c32ce1b9f9a68a15562146ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{1}}" loading="lazy">, thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\in \operatorname {Cl} _{U_{1}}(V_{1})\cap U_{2}=U_{1}\cap U_{2}\neq \emptyset }">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exists x\in \operatorname {Cl} _{U_{1}}(V_{1})\cap U_{2}=U_{1}\cap U_{2}\neq \emptyset }</annotation>

</semantics>

</math><img src="./1c931b4ee8f4bcd74ae77782db6facaeaba75757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.643ex; height:3.009ex;" alt="{\displaystyle \exists x\in \operatorname {Cl} _{U_{1}}(V_{1})\cap U_{2}=U_{1}\cap U_{2}\neq \emptyset }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6c2eaa9349c525bcf05f977029a1012c5b96462c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.509ex;" alt="{\displaystyle x\in U_{2}}" loading="lazy"> is a <a href="Point_of_closure" class="mw-redirect" title="Point of closure">point of closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./adfdbc929f16cb00bb43289c223651b41f7b9f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}" loading="lazy"> which implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}\cap U_{2}\neq \emptyset }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{1}\cap U_{2}\neq \emptyset }</annotation>

</semantics>

</math><img src="./f7ad026a6250b6dd20b5ce33b1adccaaf06ceea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.895ex; height:2.843ex;" alt="{\displaystyle V_{1}\cap U_{2}\neq \emptyset }" loading="lazy"> and a fortiori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{2}:=V\cap U_{2}\neq \emptyset }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{2}:=V\cap U_{2}\neq \emptyset }</annotation>

</semantics>

</math><img src="./7b22eeb738efae95c2e215682b4a56b9a9a41094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.427ex; height:2.843ex;" alt="{\displaystyle V_{2}:=V\cap U_{2}\neq \emptyset }" loading="lazy">. Now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V\cap (U_{1}\cup U_{2})=V_{1}\cup V_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=V\cap (U_{1}\cup U_{2})=V_{1}\cup V_{2}}</annotation>

</semantics>

</math><img src="./5a1f01d8c05ce4d13cbc0e1e1d92abdd2b82c3a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.431ex; height:2.843ex;" alt="{\displaystyle V=V\cap (U_{1}\cup U_{2})=V_{1}\cup V_{2}}" loading="lazy"> and taking the closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} _{X}(V)\supseteq {\operatorname {Cl} }_{U_{1}}(V_{1})\cup {\operatorname {Cl} }_{U_{2}}(V_{2})=U_{1}\cup U_{2}=X,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} _{X}(V)\supseteq {\operatorname {Cl} }_{U_{1}}(V_{1})\cup {\operatorname {Cl} }_{U_{2}}(V_{2})=U_{1}\cup U_{2}=X,}</annotation>

</semantics>

</math><img src="./14155806d659c3d813dbb58ac5a59fab0146fc96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.385ex; height:3.009ex;" alt="{\displaystyle \operatorname {Cl} _{X}(V)\supseteq {\operatorname {Cl} }_{U_{1}}(V_{1})\cup {\operatorname {Cl} }_{U_{2}}(V_{2})=U_{1}\cup U_{2}=X,}" loading="lazy"> therefore <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 a non-empty open and dense subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Since this is true for every non-empty open subset, <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 irreducible.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Irreducible_components">Irreducible components</h2></div>

An <a href="Irreducible_component" title="Irreducible component">irreducible component</a><a href="#cite_note-10">[10]</a> in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed.

Every irreducible subset of a space X is contained in a (not necessarily unique) irreducible component of X.<a href="#cite_note-11">[11]</a> In particular, every point of X is contained in some irreducible component of X. Unlike the <a href="Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected components</a> of a space, the irreducible components need not be disjoint (i.e. they need not form a <a href="Partition_of_a_set" title="Partition of a set">partition</a>). In general, the irreducible components will overlap.

The irreducible components of a Hausdorff space are just the <a href="Singleton_set" class="mw-redirect" title="Singleton set">singleton sets</a>.

Since every irreducible space is connected, the irreducible components will always lie in the connected components.

Every <a href="Noetherian_topological_space" title="Noetherian topological space">Noetherian topological space</a> has finitely many irreducible components.<a href="#cite_note-12">[12]</a>

<ul><li><a href="Ultraconnected_space" title="Ultraconnected space">Ultraconnected space</a></li>

<li><a href="Sober_space" title="Sober space">Sober space</a></li>

<li><a href="Geometrically_irreducible" class="mw-redirect" title="Geometrically irreducible">Geometrically irreducible</a></li></ul>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Steen & Seebach, p. 29

</li>

<li id="cite_note-FOOTNOTEHartNagataVaughan20049-2">^ <a href="#cite_ref-FOOTNOTEHartNagataVaughan20049_2-0">a</a> <a href="#cite_ref-FOOTNOTEHartNagataVaughan20049_2-1">b</a> <a href="#CITEREFHartNagataVaughan2004">Hart, Nagata & Vaughan 2004</a>, p. 9.

</li>

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</style><cite id="CITEREFVan_Douwen1993" class="citation journal cs1">Van Douwen, Eric K. (1993). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0166-8641%2893%2990147-6">"An anti-Hausdorff Fréchet space in which convergent sequences have unique limits"</a>. Topology and Its Applications. 51 (2): 147–158. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0166-8641%2893%2990147-6">10.1016/0166-8641(93)90147-6</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/004U">"Section 5.8 (004U): Irreducible components—The Stacks project"</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBourbaki1989" class="citation book cs1">Bourbaki, Nicolas (1989). Commutative Algebra: Chapters 1-7. Springer. p. 95. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64239-8</bdi>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBourbaki1989" class="citation book cs1">Bourbaki, Nicolas (1989). Commutative Algebra: Chapters 1-7. Springer. p. 95. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64239-8</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFPerrin2008" class="citation book cs1">Perrin, Daniel (2008). Algebraic Geometry. An introduction. Springer. p. 14. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-055-1</bdi>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/004W">"Lemma 5.8.3 (004W)—The Stacks project"</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBourbaki1989" class="citation book cs1">Bourbaki, Nicolas (1989). Commutative Algebra: Chapters 1-7. Springer. p. 95. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64239-8</bdi>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/004V">"Definition 5.8.1 (004V)—The Stacks project"</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/004W">"Lemma 5.8.3 (004W)—The Stacks project"</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/0050">"Section 5.9 (0050): Noetherian topological spaces—The Stacks project"</a>.</cite>

</li>

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

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

<ul><li><cite id="CITEREFHartNagataVaughan2004" class="citation book cs1">Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier/North-Holland. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-50355-8</bdi>.</cite></li>

<li><cite id="CITEREFSteenSeebach1995" class="citation cs2"><a href="Lynn_Arthur_Steen" class="mw-redirect" title="Lynn Arthur Steen">Steen, Lynn Arthur</a>; <a href="J._Arthur_Seebach%2C_Jr." class="mw-redirect" title="J. Arthur Seebach, Jr.">Seebach, J. Arthur Jr.</a> (1995) [1978], <a href="Counterexamples_in_Topology" title="Counterexamples in Topology">Counterexamples in Topology</a> (<a href="Dover_Publications" title="Dover Publications">Dover</a> reprint of 1978 ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-68735-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0507446">0507446</a></cite></li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/HyperconnectedSpace">"Hyperconnected space"</a>. <a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</cite></li></ul></div><div><div class="zim-footer">

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