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Old page wikitext, before the edit (old_wikitext) | '{{short description|Topological concept for collections of sets}}
In [[mathematics]], a collection or [[family of sets|family]] <math>\mathcal{U}</math> of subsets of a [[topological space]] <math>X</math> is said to be '''point-finite''' if every point of <math>X</math> lies in only finitely many members of <math>\mathcal{U}.</math>{{sfn|Willard|2004|p=145–152}}<ref name="w">{{citation|title=General Topology|series=Dover Books on Mathematics|first=Stephen|last=Willard|publisher=Courier Dover Publications|year=2012|pages=145–152|url=https://books.google.com/books?id=UrsHbOjiR8QC&pg=PA145|isbn=9780486131788|oclc=829161886}}.</ref>
A [[metacompact space]] is a topological space in which every [[open cover]] admits a point-finite open [[Refinement (topology)|refinement]]. Every [[locally finite collection]] of subsets of a topological space is also point-finite.
A topological space in which every open cover admits a locally finite open refinement is called a [[paracompact space]]. Every paracompact space is therefore metacompact.<ref name="w"/>
== Dieudonné's theorem ==
Since a paracompact (Hausdorff) space is normal, the next theorem applies in particular to a paracompact space.
{{math_theorem|math_statement=Let <math>X</math> be a [[normal space]]. Then each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}
==References==
{{reflist}}
*{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Une généralisation des espaces compacts | mr=0013297 | year=1944 | journal=[[Journal de Mathématiques Pures et Appliquées]]|series= Neuvième Série | issn=0021-7824 | volume=23 | pages=65–76}}
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->
* {{Willard General Topology|year=2012}} <!--{{sfn|Willard|2012|p=}}-->
{{PlanetMath attribution|id = 8398|title = point finite}}
{{topology}}
{{topology-stub}}
[[Category:General topology]]
[[Category:Families of sets]]' |
New page wikitext, after the edit (new_wikitext) | '{{short description|Topological concept for collections of sets}}
In [[mathematics]], a collection or [[family of sets|family]] <math>\mathcal{U}</math> of subsets of a [[topological space]] <math>X</math> is said to be '''point-finite''' if every point of <math>X</math> lies in only finitely many members of <math>\mathcal{U}.</math>{{sfn|Willard|2004|p=145–152}}<ref name="w">{{citation|title=General Topology|series=Dover Books on Mathematics|first=Stephen|last=Willard|publisher=Courier Dover Publications|year=2012|pages=145–152|url=https://books.google.com/books?id=UrsHbOjiR8QC&pg=PA145|isbn=9780486131788|oclc=829161886}}.</ref>
A [[metacompact space]] is a topological space in which every [[open cover]] admits a point-finite open [[Refinement (topology)|refinement]]. Every [[locally finite collection]] of subsets of a topological space is also point-finite.
A topological space in which every open cover admits a locally finite open refinement is called a [[paracompact space]]. Every paracompact space is therefore metacompact.<ref name="w"/>
== Dieudonné's theorem ==
Since a paracompact (Hausdorff) space is normal, the next theorem applies in particular to a paracompact space.
{{math_theorem|math_statement=<ref>{{harvnb|Dieudonné|1994|loc=Théorème 6.}}</ref> Let <math>X</math> be a [[normal space]]. Then each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}
==References==
{{reflist}}
*{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Une généralisation des espaces compacts | mr=0013297 | year=1944 | journal=[[Journal de Mathématiques Pures et Appliquées]]|series= Neuvième Série | issn=0021-7824 | volume=23 | pages=65–76}}
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->
* {{Willard General Topology|year=2012}} <!--{{sfn|Willard|2012|p=}}-->
{{PlanetMath attribution|id = 8398|title = point finite}}
{{topology}}
{{topology-stub}}
[[Category:General topology]]
[[Category:Families of sets]]' |
Unified diff of changes made by edit (edit_diff) | '@@ -8,5 +8,5 @@
Since a paracompact (Hausdorff) space is normal, the next theorem applies in particular to a paracompact space.
-{{math_theorem|math_statement=Let <math>X</math> be a [[normal space]]. Then each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}
+{{math_theorem|math_statement=<ref>{{harvnb|Dieudonné|1994|loc=Théorème 6.}}</ref> Let <math>X</math> be a [[normal space]]. Then each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}
==References==
' |
New page size (new_size) | 2306 |
Old page size (old_size) | 2250 |
Size change in edit (edit_delta) | 56 |
Lines added in edit (added_lines) | [
0 => '{{math_theorem|math_statement=<ref>{{harvnb|Dieudonné|1994|loc=Théorème 6.}}</ref> Let <math>X</math> be a [[normal space]]. Then each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}'
] |
Lines removed in edit (removed_lines) | [
0 => '{{math_theorem|math_statement=Let <math>X</math> be a [[normal space]]. Then each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}'
] |
Parsed HTML source of the new revision (new_html) | '<div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Topological concept for collections of sets</div>
<p>In <a href="/https/en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, a collection or <a href="/https/en.wikipedia.org/wiki/Family_of_sets" title="Family of sets">family</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">U</mi>
</mrow>
</mrow>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e63ea009de5efbca2fc285b8550daaed577c6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.038ex; width:1.635ex; height:2.176ex;" alt="{\displaystyle {\mathcal {U}}}"></span> of subsets of a <a href="/https/en.wikipedia.org/wiki/Topological_space" title="Topological space">topological space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> is said to be <b>point-finite</b> if every point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> lies in only finitely many members of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}.}">
<semantics>
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<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">U</mi>
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</mrow>
<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271d653f1fa1c669be616a0cbba39c134c275d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.038ex; width:2.282ex; height:2.176ex;" alt="{\displaystyle {\mathcal {U}}.}"></span><sup id="cite_ref-FOOTNOTEWillard2004145–152_1-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004145–152-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-w_2-0" class="reference"><a href="#cite_note-w-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>A <a href="/https/en.wikipedia.org/wiki/Metacompact_space" title="Metacompact space">metacompact space</a> is a topological space in which every <a href="/https/en.wikipedia.org/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</a> admits a point-finite open <a href="/https/en.wikipedia.org/wiki/Refinement_(topology)" class="mw-redirect" title="Refinement (topology)">refinement</a>. Every <a href="/https/en.wikipedia.org/wiki/Locally_finite_collection" title="Locally finite collection">locally finite collection</a> of subsets of a topological space is also point-finite.
A topological space in which every open cover admits a locally finite open refinement is called a <a href="/https/en.wikipedia.org/wiki/Paracompact_space" title="Paracompact space">paracompact space</a>. Every paracompact space is therefore metacompact.<sup id="cite_ref-w_2-1" class="reference"><a href="#cite_note-w-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Dieudonné's_theorem"><span id="Dieudonn.C3.A9.27s_theorem"></span>Dieudonné's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Point-finite_collection&veaction=edit&section=1" title="Edit section: Dieudonné's theorem" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Point-finite_collection&action=edit&section=1" title="Edit section's source code: Dieudonné's theorem"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Since a paracompact (Hausdorff) space is normal, the next theorem applies in particular to a paracompact space.
</p>
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<p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> be a <a href="/https/en.wikipedia.org/wiki/Normal_space" title="Normal space">normal space</a>. Then each point-finite open cover of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> has a <a href="/https/en.wikipedia.org/w/index.php?title=Shrinking_(topology)&action=edit&redlink=1" class="new" title="Shrinking (topology) (page does not exist)">shrinking</a>; that is, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{i}\mid i\in I\}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo>∣<!-- ∣ --></mo>
<mi>i</mi>
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<mi>I</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{U_{i}\mid i\in I\}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b86f693c8edae23070063ba539296067210d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.464ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\mid i\in I\}}"></span> is an open cover indexed by a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>, there is an open cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V_{i}\mid i\in I\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∣<!-- ∣ --></mo>
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mi>I</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{V_{i}\mid i\in I\}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2ff9b41b797858e859fdc839f7d987434b1952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.232ex; height:2.843ex;" alt="{\displaystyle \{V_{i}\mid i\in I\}}"></span> indexed by the same set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {V_{i}}}\subset U_{i}}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\overline {V_{i}}}\subset U_{i}}</annotation>
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}"></span>.
</p>
</div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Point-finite_collection&veaction=edit&section=2" title="Edit section: References" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Point-finite_collection&action=edit&section=2" title="Edit section's source code: References"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-FOOTNOTEWillard2004145–152-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004145–152_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 145–152.</span>
</li>
<li id="cite_note-w-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-w_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-w_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWillard2012" class="citation cs2">Willard, Stephen (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UrsHbOjiR8QC&pg=PA145"><i>General Topology</i></a>, Dover Books on Mathematics, Courier Dover Publications, pp. 145–152, <a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/9780486131788" title="Special:BookSources/9780486131788"><bdi>9780486131788</bdi></a>, <a href="/https/en.wikipedia.org/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/829161886">829161886</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.series=Dover+Books+on+Mathematics&rft.pages=145-152&rft.pub=Courier+Dover+Publications&rft.date=2012&rft_id=info%3Aoclcnum%2F829161886&rft.isbn=9780486131788&rft.aulast=Willard&rft.aufirst=Stephen&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fbooks.google.com%2Fbooks%3Fid%3DUrsHbOjiR8QC%26pg%3DPA145&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoint-finite+collection" class="Z3988"></span>.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFDieudonné1994">Dieudonné 1994</a>, Théorème 6.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFDieudonné1994 (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
</ol></div></div>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1944" class="citation cs2"><a href="/https/en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1944), "Une généralisation des espaces compacts", <i><a href="/https/en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es" title="Journal de Mathématiques Pures et Appliquées">Journal de Mathématiques Pures et Appliquées</a></i>, Neuvième Série, <b>23</b>: 65–76, <a href="/https/en.wikipedia.org/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-7824">0021-7824</a>, <a href="/https/en.wikipedia.org/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0013297">0013297</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+Math%C3%A9matiques+Pures+et+Appliqu%C3%A9es&rft.atitle=Une+g%C3%A9n%C3%A9ralisation+des+espaces+compacts&rft.volume=23&rft.pages=65-76&rft.date=1944&rft.issn=0021-7824&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0013297%23id-name%3DMR&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoint-finite+collection" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard2004" class="citation book cs1">Willard, Stephen (2004) [1970]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-o8xJQ7Ag2cC"><i>General Topology</i></a>. <a href="/https/en.wikipedia.org/wiki/Mineola,_N.Y." class="mw-redirect" title="Mineola, N.Y.">Mineola, N.Y.</a>: <a href="/https/en.wikipedia.org/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/978-0-486-43479-7" title="Special:BookSources/978-0-486-43479-7"><bdi>978-0-486-43479-7</bdi></a>. <a href="/https/en.wikipedia.org/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/115240">115240</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=2004&rft_id=info%3Aoclcnum%2F115240&rft.isbn=978-0-486-43479-7&rft.aulast=Willard&rft.aufirst=Stephen&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fbooks.google.com%2Fbooks%3Fid%3D-o8xJQ7Ag2cC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoint-finite+collection" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard2012" class="citation book cs1">Willard, Stephen (2012) [1970]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UrsHbOjiR8QC"><i>General Topology</i></a>. <a href="/https/en.wikipedia.org/wiki/Mineola,_N.Y." class="mw-redirect" title="Mineola, N.Y.">Mineola, N.Y.</a>: Courier <a href="/https/en.wikipedia.org/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/9780486131788" title="Special:BookSources/9780486131788"><bdi>9780486131788</bdi></a>. <a href="/https/en.wikipedia.org/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/829161886">829161886</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.place=Mineola%2C+N.Y.&rft.pub=Courier+Dover+Publications&rft.date=2012&rft_id=info%3Aoclcnum%2F829161886&rft.isbn=9780486131788&rft.aulast=Willard&rft.aufirst=Stephen&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fbooks.google.com%2Fbooks%3Fid%3DUrsHbOjiR8QC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoint-finite+collection" class="Z3988"></span></li></ul>
<p><i>This article incorporates material from point finite on <a href="/https/en.wikipedia.org/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/https/en.wikipedia.org/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i>
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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="/https/en.wikipedia.org/wiki/General_topology" title="General topology">General (point-set)</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Continuum_(topology)" title="Continuum (topology)">Continuum</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Differential_topology" title="Differential topology">Differential</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Geometric_topology" title="Geometric topology">Geometric</a>
<ul><li><a href="/https/en.wikipedia.org/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li>
<li><a href="/https/en.wikipedia.org/wiki/Homology_(mathematics)" title="Homology (mathematics)">Homology</a>
<ul><li><a href="/https/en.wikipedia.org/wiki/Cohomology" title="Cohomology">cohomology</a></li></ul></li>
<li><a href="/https/en.wikipedia.org/wiki/Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>
<li><a href="/https/en.wikipedia.org/wiki/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><span typeof="mw:File"><a href="/https/en.wikipedia.org/wiki/Klein_bottle" title="Klein bottle"><img alt="Computer graphics rendering of a Klein bottle" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/60px-Kleinsche_Flasche.png" decoding="async" width="60" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/90px-Kleinsche_Flasche.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/120px-Kleinsche_Flasche.png 2x" data-file-width="1171" data-file-height="1561" /></a></span></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="/https/en.wikipedia.org/wiki/Open_set" title="Open set">Open set</a> / <a href="/https/en.wikipedia.org/wiki/Closed_set" title="Closed set">Closed set</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Topological_space" title="Topological space">Space</a>
<ul><li><a href="/https/en.wikipedia.org/wiki/Compact_space" title="Compact space">compact</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Connected_space" title="Connected space">connected</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Metric_space" title="Metric space">metric</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Uniform_space" title="Uniform space">uniform</a></li></ul></li>
<li><a href="/https/en.wikipedia.org/wiki/Homotopy" title="Homotopy">Homotopy</a>
<ul><li><a href="/https/en.wikipedia.org/wiki/Homotopy_group" title="Homotopy group">homotopy group</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li>
<li><a href="/https/en.wikipedia.org/wiki/Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li>
<li><a href="/https/en.wikipedia.org/wiki/CW_complex" title="CW complex">CW complex</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Manifold" title="Manifold">Manifold</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Second-countable_space" title="Second-countable space">Second-countable space</a></li>
<li><a href="/https/en.wikipedia.org/wiki/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="/https/en.wikipedia.org/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Betti_number" title="Betti number">Betti number</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Winding_number" title="Winding number">Winding number</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Chern_class" title="Chern class">Chern number</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Orientability" title="Orientability">Orientability</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;"><a href="/https/en.wikipedia.org/wiki/Category:Theorems_in_topology" title="Category:Theorems in topology">Key results</a></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="/https/en.wikipedia.org/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li>
<li><a href="/https/en.wikipedia.org/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/https/en.wikipedia.org/wiki/Category:Topology" title="Category:Topology">Category</a></li>
<li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/https/en.wikipedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/https/en.wikipedia.org/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li>
<li><span class="noviewer" typeof="mw:File"><a href="/https/en.wikipedia.org/wiki/File:Wikibooks-logo.svg" class="mw-file-description" title="Wikibooks page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> <a href="https://en.wikibooks.org/wiki/Topology" class="extiw" title="wikibooks:Topology">Wikibook</a></li>
<li><span class="noviewer" typeof="mw:File"><a href="/https/en.wikipedia.org/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description" title="Wikiversity page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw" title="wikiversity:Topology">Wikiversity</a></li>
<li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/https/en.wikipedia.org/wiki/List_of_topology_topics" title="List of topology topics">Topics</a>
<ul><li><a href="/https/en.wikipedia.org/wiki/List_of_general_topology_topics" title="List of general topology topics">general</a></li>
<li><a href="/https/en.wikipedia.org/wiki/List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li>
<li><a href="/https/en.wikipedia.org/wiki/List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li>
<li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/https/en.wikipedia.org/wiki/List_of_important_publications_in_mathematics#Topology" title="List of important publications in mathematics">Publications</a></li></ul>
</div></td></tr></tbody></table></div>
<style data-mw-deduplicate="TemplateStyles:r1012311289">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/https/en.wikipedia.org/wiki/File:KleinBottle-01.png" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/KleinBottle-01.png/30px-KleinBottle-01.png" decoding="async" width="30" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/KleinBottle-01.png/45px-KleinBottle-01.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/KleinBottle-01.png/60px-KleinBottle-01.png 2x" data-file-width="240" data-file-height="300" /></a></span></td><td><p class="asbox-body">This <a href="/https/en.wikipedia.org/wiki/Topology" title="Topology">topology-related</a> article is a <a href="/https/en.wikipedia.org/wiki/Wikipedia:Stub" title="Wikipedia:Stub">stub</a>. You can help Wikipedia by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Point-finite_collection&action=edit">expanding it</a>.</p></td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/https/en.wikipedia.org/wiki/Template:Topology-stub" title="Template:Topology-stub"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/https/en.wikipedia.org/wiki/Template_talk:Topology-stub" title="Template talk:Topology-stub"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/https/en.wikipedia.org/wiki/Special:EditPage/Template:Topology-stub" title="Special:EditPage/Template:Topology-stub"><abbr title="Edit this template">e</abbr></a></li></ul></div></div></div>' |
Whether or not the change was made through a Tor exit node (tor_exit_node) | false |
Unix timestamp of change (timestamp) | '1731836614' |