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Является центральным понятием <a href="/wiki/%D0%9E%D0%B1%D1%89%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Общая топология">общей топологии</a>. </p><p>Наряду с понятием <a href="/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрическое пространство">метрического пространства</a>, является одной из разновидностей пространств в <a href="/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Геометрия">геометрии</a>. В топологических пространствах не определены понятия <a href="/wiki/%D0%A0%D0%B0%D1%81%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Расстояние">расстояний</a>, величин <a href="/wiki/%D0%A3%D0%B3%D0%BE%D0%BB" title="Угол">углов</a>, <a href="/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0%D0%B4%D1%8C" title="Площадь">площадей</a> и <a href="/wiki/%D0%9E%D0%B1%D1%8A%D1%91%D0%BC" title="Объём">объёмов</a>, но возможно говорить о <a href="/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Непрерывное отображение">непрерывности</a>, <a href="/wiki/%D0%9F%D1%80%D0%B5%D0%B4%D0%B5%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Предел (математика)">сходимости</a> и <a href="/wiki/%D0%A1%D0%B2%D1%8F%D0%B7%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Связное пространство">связности</a>. Для этого в них определяется качественное (в отличие от количественного) понятие близости элементов. </p><p>Типичными топологическими пространствами являются <a href="/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Евклидово пространство">евклидовы пространства</a> и их подпространства, <a href="/wiki/%D0%A8%D0%B0%D1%80" title="Шар">шары</a> и <a href="/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D1%81%D1%84%D0%B5%D1%80%D0%B0" title="Гиперсфера">сферы</a>, графы и произвольные <a href="/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B8%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплициальный комплекс">симплициальные</a> и <a href="/wiki/CW-%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="CW-комплекс">CW-комплексы</a>, а также <a href="/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Поверхность">поверхности</a> и <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Многообразие">многообразия</a> произвольной размерности. </p><p>Каждое метрическое пространство естественным образом индуцирует топологическую структуру, но разные метрические пространства могут задавать одинаковые топологические. Кроме того, современное понятие топологического пространства допускает <a href="/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%B7%D1%83%D0%B5%D0%BC%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метризуемое пространство">неметризуемые</a> пространства, то есть такие, которые не могут быть получены из метрических. </p><p>Понятие топологического пространства позволяет привнести геометрические образы в любую область математики, как бы далека от геометрии эта область ни была на первый взгляд. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="ru" dir="ltr"><h2 id="mw-toc-heading">Содержание</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Определение"><span class="tocnumber">1</span> <span class="toctext">Определение</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Дополнительные_аксиомы"><span class="tocnumber">1.1</span> <span class="toctext">Дополнительные аксиомы</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#Примеры"><span class="tocnumber">2</span> <span class="toctext">Примеры</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Способы_задания_топологии"><span class="tocnumber">3</span> <span class="toctext">Способы задания топологии</span></a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Задание_топологии_с_помощью_базы_или_предбазы"><span class="tocnumber">3.1</span> <span class="toctext">Задание топологии с помощью базы или предбазы</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Индуцированная_топология"><span class="tocnumber">3.2</span> <span class="toctext">Индуцированная топология</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Фактортопология"><span class="tocnumber">3.3</span> <span class="toctext">Фактортопология</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Задание_топологии_с_помощью_замкнутых_множеств"><span class="tocnumber">3.4</span> <span class="toctext">Задание топологии с помощью замкнутых множеств</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Непрерывные_отображения"><span class="tocnumber">4</span> <span class="toctext">Непрерывные отображения</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Функциональная_структура"><span class="tocnumber">5</span> <span class="toctext">Функциональная структура</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#См._также"><span class="tocnumber">6</span> <span class="toctext">См. также</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Примечания"><span class="tocnumber">7</span> <span class="toctext">Примечания</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Литература"><span class="tocnumber">8</span> <span class="toctext">Литература</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Определение"><span id=".D0.9E.D0.BF.D1.80.D0.B5.D0.B4.D0.B5.D0.BB.D0.B5.D0.BD.D0.B8.D0.B5"></span>Определение</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=1" title="Редактировать раздел «Определение»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=1" title="Редактировать код раздела «Определение»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть дано <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество">множество</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> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </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>. Система <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> его подмножеств называется <i><b>тополо́гией</b></i> на <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> </semantics> </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>, если выполнены следующие условия: </p> <ol><li><a href="/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D0%B8_%D0%BD%D0%B0%D0%B4_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D0%BC%D0%B8" class="mw-redirect" title="Операции над множествами">Объединение</a> произвольного семейства множеств, принадлежащих <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>, принадлежит <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>; то есть для любого индексирующего множества <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> и семейства <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_{\alpha }\in {\mathcal {T}},\alpha \in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\alpha }\in {\mathcal {T}},\alpha \in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d77953779fb7d17742bac0bd7d547d614437af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.754ex; height:2.676ex;" alt="{\displaystyle U_{\alpha }\in {\mathcal {T}},\alpha \in A}"></span>, выполнено <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 \bigcup \limits _{\alpha \in A}U_{\alpha }\in {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \limits _{\alpha \in A}U_{\alpha }\in {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b3fa174b6e35f9c5ce3ea252fe89b95d0b1aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.416ex; height:5.676ex;" alt="{\displaystyle \bigcup \limits _{\alpha \in A}U_{\alpha }\in {\mathcal {T}}}"></span>.</li> <li><a href="/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D0%B8_%D0%BD%D0%B0%D0%B4_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D0%BC%D0%B8" class="mw-redirect" title="Операции над множествами">Пересечение</a> конечного семейства множеств, принадлежащих <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>, принадлежит <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>; то есть если <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}\in {\mathcal {T}}\quad (i=1,\;\ldots ,\;n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}\in {\mathcal {T}}\quad (i=1,\;\ldots ,\;n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a0ab45b03d9e32649bc84e89d732d8596263d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.222ex; height:2.843ex;" alt="{\displaystyle U_{i}\in {\mathcal {T}}\quad (i=1,\;\ldots ,\;n)}"></span>, то <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 \bigcap \limits _{i=1}^{n}U_{i}\in {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo movablelimits="false">⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcap \limits _{i=1}^{n}U_{i}\in {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c0ba96e461353f2994f3017162e5e5a29b0474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.219ex; height:6.843ex;" alt="{\displaystyle \bigcap \limits _{i=1}^{n}U_{i}\in {\mathcal {T}}}"></span>.</li> <li><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,\;\varnothing \in {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi class="MJX-variant">∅</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,\;\varnothing \in {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86f4c8b01988d72100d925d23ec3d604b7b5012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.244ex; height:2.676ex;" alt="{\displaystyle X,\;\varnothing \in {\mathcal {T}}}"></span>.</li></ol> <p>Пара <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,\;{\mathcal {T}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\;{\mathcal {T}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155075cd3380aba57315ae58e865833373a052ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.404ex; height:2.843ex;" alt="{\displaystyle (X,\;{\mathcal {T}})}"></span> называется <b>топологическим пространством</b>. Множества, принадлежащие <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>, называются <i><b><a href="/wiki/%D0%9E%D1%82%D0%BA%D1%80%D1%8B%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Открытое множество">открытыми множествами</a></b></i>. </p><p>Множества, являющиеся дополнениями к открытым, называются <a href="/wiki/%D0%97%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Замкнутое множество">замкнутыми</a>. </p><p>Всякое открытое множество, содержащее данную точку, называется её <a href="/wiki/%D0%9E%D0%BA%D1%80%D0%B5%D1%81%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окрестность">окрестностью</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Дополнительные_аксиомы"><span id=".D0.94.D0.BE.D0.BF.D0.BE.D0.BB.D0.BD.D0.B8.D1.82.D0.B5.D0.BB.D1.8C.D0.BD.D1.8B.D0.B5_.D0.B0.D0.BA.D1.81.D0.B8.D0.BE.D0.BC.D1.8B"></span>Дополнительные аксиомы</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=2" title="Редактировать раздел «Дополнительные аксиомы»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=2" title="Редактировать код раздела «Дополнительные аксиомы»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Три аксиомы, определяющие общий класс топологических пространств, часто дополняются теми или иными <a href="/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D1%8B_%D0%BE%D1%82%D0%B4%D0%B5%D0%BB%D0%B8%D0%BC%D0%BE%D1%81%D1%82%D0%B8" title="Аксиомы отделимости">аксиомами отделимости</a>, в зависимости от которых выделяют различные классы топологических пространств, например, тихоновские пространства, <a href="/wiki/%D0%A5%D0%B0%D1%83%D1%81%D0%B4%D0%BE%D1%80%D1%84%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Хаусдорфово пространство">хаусдорфовы пространства</a>, регулярные, вполне регулярные, нормальные пространства и др. </p><p>Кроме этого, на свойства топологических пространств сильно влияет выполнение тех или иных аксиом счётности — <a href="/wiki/%D0%9F%D0%B5%D1%80%D0%B2%D0%B0%D1%8F_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0_%D1%81%D1%87%D1%91%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Первая аксиома счётности">первая аксиома счётности</a>, <a href="/wiki/%D0%92%D1%82%D0%BE%D1%80%D0%B0%D1%8F_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0_%D1%81%D1%87%D1%91%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Вторая аксиома счётности">вторая аксиома счётности</a> (пространства со счётной базой топологии), а также <a href="/wiki/%D0%A1%D0%B5%D0%BF%D0%B0%D1%80%D0%B0%D0%B1%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" class="mw-redirect" title="Сепарабельность">сепарабельность</a> пространства. Из наличия счётной базы топологии следует сепарабельность и выполнение первой аксиомы счётности. Кроме того, например, регулярные пространства со счётной базой являются нормальными и, более того, метризуемы, то есть их топология может быть задана некоторой метрикой. Для компактных хаусдорфовых пространств наличие счётной базы топологии является необходимым и достаточным условием метризуемости. Для метрических пространств наличие счётной базы топологии и сепарабельность — эквивалентны. </p> <div class="mw-heading mw-heading2"><h2 id="Примеры"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.80.D1.8B"></span>Примеры</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=3" title="Редактировать раздел «Примеры»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=3" title="Редактировать код раздела «Примеры»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%D0%A1%D0%B2%D1%8F%D0%B7%D0%BD%D0%BE%D0%B5_%D0%B4%D0%B2%D0%BE%D0%B5%D1%82%D0%BE%D1%87%D0%B8%D0%B5" title="Связное двоеточие">Связное двоеточие</a> — двуточечное топологическое пространство. </p><p>Вещественная прямая <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }"></span> является топологическим пространством, если, например, назвать открытыми множествами произвольные (пустые, конечные или бесконечные) объединения конечных или бесконечных интервалов. Множество всех конечных открытых интервалов <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 \{(a,\;b)\mid a,\;b\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>a</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(a,\;b)\mid a,\;b\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afc6748fe7901cd5af250bc23fdcd75bfea1e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.403ex; height:2.843ex;" alt="{\displaystyle \{(a,\;b)\mid a,\;b\in \mathbb {R} \}}"></span> является <a href="/wiki/%D0%91%D0%B0%D0%B7%D0%B0_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B8" title="База топологии">базой этой топологии</a>. Это — стандартная топология на прямой. Вообще же на множестве вещественных чисел можно ввести очень разнообразные топологии, например, <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 \mathbb {R} _{\to }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\to }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ec33ac38b2739547f50de7a4bf4f314e80288c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.553ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{\to }}"></span>, прямая с «топологией стрелки», где открытые множества имеют вид <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 (a,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e650c4e1b70a57c49b18df445fc494d0849e133b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" alt="{\displaystyle (a,\infty )}"></span>, или <a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F_%D0%97%D0%B0%D1%80%D0%B8%D1%81%D1%81%D0%BA%D0%BE%D0%B3%D0%BE" title="Топология Зарисского">топология Зарисского</a>, в которой любое замкнутое множество — это конечное множество точек. </p><p>Вообще, <a href="/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Евклидово пространство">евклидовы пространства</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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> являются топологическими пространствами. Базой их стандартной топологии можно выбрать открытые шары или открытые кубы. Обобщая далее, всякое <a href="/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрическое пространство">метрическое пространство</a> является топологическим пространством, базу топологии которого составляют <a href="/wiki/%D0%9E%D1%82%D0%BA%D1%80%D1%8B%D1%82%D1%8B%D0%B9_%D1%88%D0%B0%D1%80" class="mw-redirect" title="Открытый шар">открытые шары</a>. Таковы, например, изучаемые в <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Функциональный анализ">функциональном анализе</a> бесконечномерные пространства функций. </p><p>Множество <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 C(X,\;Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(X,\;Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05d3df2ffca4ff551ffa4be49e5c762b7d50915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.008ex; height:2.843ex;" alt="{\displaystyle C(X,\;Y)}"></span> непрерывных отображений топологического пространства <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> </semantics> </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> в топологическое пространство <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> является топологическим пространством относительно следующей топологии, которая называется <i>компактно-открытой</i>. Предбаза задаётся множествами <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 C(K,\;U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K,\;U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d13b6fa68334b85ef25642dc24faaf52f02ed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.103ex; height:2.843ex;" alt="{\displaystyle C(K,\;U)}"></span>, состоящими из отображений, при которых образ компакта <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> в <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> </semantics> </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> лежит в открытом множестве <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}"></span> в <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>. </p><p>Произвольное множество <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> </semantics> </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> можно сделать топологическим пространством, если называть открытыми все его подмножества. Такая топология называется <a href="/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" class="mw-redirect" title="Дискретная топология">дискретной</a>. В ней любые множества являются открытыми. Другой предельный случай — назвать открытыми минимально возможное количество подмножеств <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> </semantics> </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>, а именно, ввести <a href="/wiki/%D0%A2%D1%80%D0%B8%D0%B2%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Тривиальная топология">тривиальную топологию</a> — в ней открытыми являются лишь пустое множество и само пространство <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> </semantics> </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>. </p> <div class="mw-heading mw-heading2"><h2 id="Способы_задания_топологии"><span id=".D0.A1.D0.BF.D0.BE.D1.81.D0.BE.D0.B1.D1.8B_.D0.B7.D0.B0.D0.B4.D0.B0.D0.BD.D0.B8.D1.8F_.D1.82.D0.BE.D0.BF.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D0.B8"></span>Способы задания топологии</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=4" title="Редактировать раздел «Способы задания топологии»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=4" title="Редактировать код раздела «Способы задания топологии»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Задание_топологии_с_помощью_базы_или_предбазы"><span id=".D0.97.D0.B0.D0.B4.D0.B0.D0.BD.D0.B8.D0.B5_.D1.82.D0.BE.D0.BF.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D0.B8_.D1.81_.D0.BF.D0.BE.D0.BC.D0.BE.D1.89.D1.8C.D1.8E_.D0.B1.D0.B0.D0.B7.D1.8B_.D0.B8.D0.BB.D0.B8_.D0.BF.D1.80.D0.B5.D0.B4.D0.B1.D0.B0.D0.B7.D1.8B"></span>Задание топологии с помощью базы или предбазы</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=5" title="Редактировать раздел «Задание топологии с помощью базы или предбазы»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=5" title="Редактировать код раздела «Задание топологии с помощью базы или предбазы»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote navigation-not-searchable">Основная статья: <b><a href="/wiki/%D0%91%D0%B0%D0%B7%D0%B0_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B8" title="База топологии">База топологии</a></b></div> <p>Не всегда удобно перечислять все открытые множества. Часто удобнее указать некоторый меньший набор открытых множеств, который порождает их все. Формализацией этого является понятие базы топологии. Подмножество топологии <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 {\mathfrak {B}}\subset {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}\subset {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f5680e4db4041464e88484d6dd71dd220ed57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.089ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {B}}\subset {\mathcal {T}}}"></span> называется <a href="/wiki/%D0%91%D0%B0%D0%B7%D0%B0_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B8" title="База топологии">базой топологии</a>, если всякое открытое множество представляется как объединение множеств из <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 {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span>, то есть </p> <dl><dd><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 \forall U\in {\mathcal {T}}\;\exists \{U_{\alpha }\}_{\alpha \in A}\subset {\mathfrak {B}}\colon U=\bigcup \limits _{\alpha \in A}U_{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>U</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mo>:</mo> <mi>U</mi> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall U\in {\mathcal {T}}\;\exists \{U_{\alpha }\}_{\alpha \in A}\subset {\mathfrak {B}}\colon U=\bigcup \limits _{\alpha \in A}U_{\alpha }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e034b7af2e8bd91897541179ed2e9229c43e02b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.953ex; height:5.676ex;" alt="{\displaystyle \forall U\in {\mathcal {T}}\;\exists \{U_{\alpha }\}_{\alpha \in A}\subset {\mathfrak {B}}\colon U=\bigcup \limits _{\alpha \in A}U_{\alpha }.}"></span></dd></dl> <p>Ещё более экономный способ задания топологии состоит в задании её <a href="/wiki/%D0%9F%D1%80%D0%B5%D0%B4%D0%B1%D0%B0%D0%B7%D0%B0" class="mw-redirect" title="Предбаза">предбазы</a> — множества, которое становится базой, если к нему прибавить произвольные конечные пересечения его элементов. Для того, чтобы систему множеств <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 {\mathfrak {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f306de895b1fd787a364c508badc106b1ff73e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}"></span> можно было объявить предбазой топологии, необходимо и достаточно, чтобы она покрывала всё множество <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> </semantics> </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>. </p><p>Наиболее часто предбазы используются для задания топологии, индуцированной на <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> </semantics> </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> семейством отображений (см. далее). </p> <div class="thumb tright" style="position:relative; left:-1px; width:312px;"><div class="thumbinner"><div class="thumbimage" style="margin:1px; width:300px; overflow:hidden;"><div style="margin:-0px -0px -290px -0px;"><span typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Refinement_on_a_planar_shape.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Refinement_on_a_planar_shape.svg/300px-Refinement_on_a_planar_shape.svg.png" decoding="async" width="300" height="585" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Refinement_on_a_planar_shape.svg/450px-Refinement_on_a_planar_shape.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Refinement_on_a_planar_shape.svg/600px-Refinement_on_a_planar_shape.svg.png 2x" data-file-width="378" data-file-height="737" /></a></span></div></div><div class="thumbcaption" style="clear:left;text-align:left;background:transparent;"><div class="magnify noprint" style="float: right; margin:-0px -0px -0px +5px;"><span typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Refinement_on_a_planar_shape.svg" title="Файл:Refinement on a planar shape.svg"><img src="//upload.wikimedia.org/wikipedia/commons/6/6b/Magnify-clip.png" decoding="async" width="15" height="11" class="mw-file-element" data-file-width="15" data-file-height="11" /></a></span></div>Конструкция индуцированной топологии с плоскости на квадрат (чёрный) подразумевает, что открытыми подмножествами квадрата объявляются его пересечения с открытыми подмножествами плоскости.</div></div></div> <div class="mw-heading mw-heading3"><h3 id="Индуцированная_топология"><span id=".D0.98.D0.BD.D0.B4.D1.83.D1.86.D0.B8.D1.80.D0.BE.D0.B2.D0.B0.D0.BD.D0.BD.D0.B0.D1.8F_.D1.82.D0.BE.D0.BF.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D1.8F"></span>Индуцированная топология</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=6" title="Редактировать раздел «Индуцированная топология»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=6" title="Редактировать код раздела «Индуцированная топология»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote navigation-not-searchable">Основная статья: <b><a href="/wiki/%D0%98%D0%BD%D0%B4%D1%83%D1%86%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Индуцированная топология">Индуцированная топология</a></b></div> <p>Пусть <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 f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> — произвольное отображение множества <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> </semantics> </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> в топологическое пространство <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>. Индуцированная топология даёт естественный способ введения топологии на <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> </semantics> </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>: за открытые множества в <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> </semantics> </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> берутся всевозможные прообразы открытых множеств в <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>; то есть <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\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f323e60081af33e440424cb4e11372bb4a196043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.603ex; height:2.176ex;" alt="{\displaystyle U\in X}"></span> открыто, если существует открытое <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\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fb457f2702545db3b3dce45a14c9a80a67320c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.401ex; height:2.176ex;" alt="{\displaystyle V\in Y}"></span> такое, что <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=f^{-1}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=f^{-1}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8644a26c59e478fd08c40db2f06458e6e7165b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.131ex; height:3.176ex;" alt="{\displaystyle U=f^{-1}(V)}"></span>. Топология на <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> </semantics> </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>, описанная выше, является минимальной и единственной (по включению) топологией, в которой данное отображение является непрерывным. </p><p><b>Пример.</b> Пусть <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> </semantics> </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> топологическое пространство, <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> его подмножество. Если применить описанную выше конструкцию к теоретико-множественному вложению <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:A\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97059160d1002162c22e1f2f5e4c2aee2afaf629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.077ex; height:2.176ex;" alt="{\displaystyle i:A\to X}"></span>, то получим топологию на подмножестве, обычно называемую также индуцированной. </p> <div class="mw-heading mw-heading3"><h3 id="Фактортопология"><span id=".D0.A4.D0.B0.D0.BA.D1.82.D0.BE.D1.80.D1.82.D0.BE.D0.BF.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D1.8F"></span>Фактортопология</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=7" title="Редактировать раздел «Фактортопология»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=7" title="Редактировать код раздела «Фактортопология»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть <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> </semantics> </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> — топологическое пространство, пусть также на нём задано некоторое <a href="/wiki/%D0%9E%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5_%D1%8D%D0%BA%D0%B2%D0%B8%D0%B2%D0%B0%D0%BB%D0%B5%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Отношение эквивалентности">отношение эквивалентности</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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span>, в таком случае есть естественный способ задать топологию на фактормножестве <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/{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80508302af0d1ad574ebaaecbfd8b553e88149d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.951ex; height:2.843ex;" alt="{\displaystyle X/{\sim }}"></span>. Мы объявляем подмножество фактора открытым тогда и только тогда, когда его прообраз при отображении факторизации является открытым в <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> </semantics> </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>. Легко проверить, во-первых, что это действительно определяет топологию, во-вторых, что это максимальная и единственная (по включению) топология, в которой указанное отображение факторизации непрерывно. Такая топология обычно называется фактортопологией на <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/{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80508302af0d1ad574ebaaecbfd8b553e88149d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.951ex; height:2.843ex;" alt="{\displaystyle X/{\sim }}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Задание_топологии_с_помощью_замкнутых_множеств"><span id=".D0.97.D0.B0.D0.B4.D0.B0.D0.BD.D0.B8.D0.B5_.D1.82.D0.BE.D0.BF.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D0.B8_.D1.81_.D0.BF.D0.BE.D0.BC.D0.BE.D1.89.D1.8C.D1.8E_.D0.B7.D0.B0.D0.BC.D0.BA.D0.BD.D1.83.D1.82.D1.8B.D1.85_.D0.BC.D0.BD.D0.BE.D0.B6.D0.B5.D1.81.D1.82.D0.B2"></span>Задание топологии с помощью замкнутых множеств</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=8" title="Редактировать раздел «Задание топологии с помощью замкнутых множеств»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=8" title="Редактировать код раздела «Задание топологии с помощью замкнутых множеств»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Множество <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 F\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253df0b206fc9c6fa0e67b94d7eb807e9f253274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.819ex; height:2.176ex;" alt="{\displaystyle F\subset X}"></span> называется <a href="/wiki/%D0%97%D0%B0%D0%BC%D0%BA%D0%BD%D1%83%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Замкнутое множество">замкнутым</a>, если его <a href="/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D0%B8_%D0%BD%D0%B0%D0%B4_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D0%BC%D0%B8" class="mw-redirect" title="Операции над множествами">дополнение</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 U=X\setminus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>X</mi> <mo class="MJX-variant">∖</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=X\setminus F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d36e124e80ad93a477cd69fe513f14a4e4b9d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.797ex; height:2.843ex;" alt="{\displaystyle U=X\setminus F}"></span> — открытое множество. Задать топологию на <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> </semantics> </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> системой замкнутых множеств — значит предъявить систему <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 {P}}}"> <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">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span> подмножеств <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> </semantics> </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> со свойствами: </p> <ol><li>Система <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 {P}}}"> <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">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span> замкнута относительно операции пересечения множеств (в том числе бесконечных семейств): <dl><dd><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 \forall \alpha \in A\quad F_{\alpha }\in {\mathcal {P}}\Rightarrow \bigcap \limits _{\alpha \in A}F_{\alpha }\in {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>∈</mo> <mi>A</mi> <mspace width="1em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">⇒</mo> <munder> <mo movablelimits="false">⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \alpha \in A\quad F_{\alpha }\in {\mathcal {P}}\Rightarrow \bigcap \limits _{\alpha \in A}F_{\alpha }\in {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7270b56e6d84e2c1743d1bb67ed68a40545917da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.714ex; height:5.676ex;" alt="{\displaystyle \forall \alpha \in A\quad F_{\alpha }\in {\mathcal {P}}\Rightarrow \bigcap \limits _{\alpha \in A}F_{\alpha }\in {\mathcal {P}}}"></span></dd></dl></li> <li>Система <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 {P}}}"> <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">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span> замкнута относительно операции объединения множеств (в конечном количестве): <dl><dd><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 F_{1},\;F_{2}\in {\mathcal {P}}\Rightarrow F_{1}\cup F_{2}\in {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∪</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1},\;F_{2}\in {\mathcal {P}}\Rightarrow F_{1}\cup F_{2}\in {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab2d6f88779ac3713fb9282c774abefb21e83c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.16ex; height:2.509ex;" alt="{\displaystyle F_{1},\;F_{2}\in {\mathcal {P}}\Rightarrow F_{1}\cup F_{2}\in {\mathcal {P}}}"></span></dd></dl></li> <li>Множества <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,\;\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,\;\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aaba50b5c5767752aeb0b11be505f7215382fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.467ex; height:2.509ex;" alt="{\displaystyle X,\;\varnothing }"></span> включены в систему <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 {P}}}"> <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">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>.</li></ol> <p>Если система множеств с такими свойствами задана, с помощью операции дополнения строится система <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>открытых множеств, задающая топологию на <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> </semantics> </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>. </p> <dl><dd><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 {T}}=\{X\setminus F:F\in {\mathcal {P}}\}.}"> <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">T</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo class="MJX-variant">∖</mo> <mi>F</mi> <mo>:</mo> <mi>F</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}=\{X\setminus F:F\in {\mathcal {P}}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921e022dd7544d31dd36b9aaf33774a171898c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.144ex; height:2.843ex;" alt="{\displaystyle {\mathcal {T}}=\{X\setminus F:F\in {\mathcal {P}}\}.}"></span></dd></dl> <p>В <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебраическая геометрия">алгебраической геометрии</a> применяется задание топологии на <a href="/wiki/%D0%A1%D0%BF%D0%B5%D0%BA%D1%82%D1%80_%D0%BA%D0%BE%D0%BB%D1%8C%D1%86%D0%B0" title="Спектр кольца">спектре</a> (системе всех <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE%D0%B9_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB" title="Простой идеал">простых идеалов</a>) коммутативного кольца с единицей <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> — <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=\mathrm {Spec} \,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> <mspace width="thinmathspace" /> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathrm {Spec} \,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c19d7a2ae2d275e0cd028912a83aa810d1bd79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.879ex; height:2.509ex;" alt="{\displaystyle X=\mathrm {Spec} \,B}"></span>. Топология на <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> </semantics> </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> вводится с помощью системы замкнутых множеств: пусть <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 {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"></span> — произвольный <a href="/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Идеал (алгебра)">идеал</a> кольца <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> (не обязательно простой), тогда ему соответствует множество </p> <dl><dd><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({\mathfrak {a}})=\{{\mathfrak {p}}\in X:{\mathfrak {a}}\subset {\mathfrak {p}}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V({\mathfrak {a}})=\{{\mathfrak {p}}\in X:{\mathfrak {a}}\subset {\mathfrak {p}}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0e63698a4bc932844b375a1df43e4d349b4a8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.173ex; height:2.843ex;" alt="{\displaystyle V({\mathfrak {a}})=\{{\mathfrak {p}}\in X:{\mathfrak {a}}\subset {\mathfrak {p}}\}.}"></span></dd></dl> <p>Все множества такого вида образуют систему множеств, удовлетворяющую перечисленным аксиомам, так как </p> <dl><dd><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 \bigcap \limits _{\alpha \in A}V({\mathfrak {a}}_{\alpha })=V\left(\sum \limits _{\alpha \in A}{\mathfrak {a}}_{\alpha }\right),\quad V({\mathfrak {a}})\cup V({\mathfrak {b}})=V({\mathfrak {a}}\cdot {\mathfrak {b}}),\quad V((0))=X,\quad V((1))=\varnothing .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>∪</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>V</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mo>,</mo> <mspace width="1em" /> <mi>V</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcap \limits _{\alpha \in A}V({\mathfrak {a}}_{\alpha })=V\left(\sum \limits _{\alpha \in A}{\mathfrak {a}}_{\alpha }\right),\quad V({\mathfrak {a}})\cup V({\mathfrak {b}})=V({\mathfrak {a}}\cdot {\mathfrak {b}}),\quad V((0))=X,\quad V((1))=\varnothing .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2868386e29b7110b40ffbc7cabcf8b157cc2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:82.065ex; height:7.509ex;" alt="{\displaystyle \bigcap \limits _{\alpha \in A}V({\mathfrak {a}}_{\alpha })=V\left(\sum \limits _{\alpha \in A}{\mathfrak {a}}_{\alpha }\right),\quad V({\mathfrak {a}})\cup V({\mathfrak {b}})=V({\mathfrak {a}}\cdot {\mathfrak {b}}),\quad V((0))=X,\quad V((1))=\varnothing .}"></span></dd></dl> <p><a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F_%D0%97%D0%B0%D1%80%D0%B8%D1%81%D1%81%D0%BA%D0%BE%D0%B3%D0%BE" title="Топология Зарисского">Топология Зарисского</a> в пространстве <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=\mathbf {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbf {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748f277e18fc2e610556425f54f51595d3fa19de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.228ex; height:2.343ex;" alt="{\displaystyle X=\mathbf {C} ^{n}}"></span> также задаётся с помощью системы замкнутых множеств. Замкнутыми множествами в топологии Зарисского принимаются все множества, являющиеся множеством общих нулей конечной системы многочленов. Выполнение аксиом системы замкнутых множеств следует из <a href="/wiki/%D0%9D%D1%91%D1%82%D0%B5%D1%80%D0%BE%D0%B2%D0%BE_%D0%BA%D0%BE%D0%BB%D1%8C%D1%86%D0%BE" title="Нётерово кольцо">нётеровости</a> <a href="/wiki/%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD%D0%BE%D0%B2" title="Кольцо многочленов">кольца многочленов</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 \mathbf {C} [z_{1},\;z_{2},\;\ldots ,\;z_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} [z_{1},\;z_{2},\;\ldots ,\;z_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a05290f039a7b40c0a3df81308334f5829f93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.943ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} [z_{1},\;z_{2},\;\ldots ,\;z_{n}]}"></span> и того факта, что общие нули произвольной системы многочленов совпадают с общими нулями идеала, который они образуют. </p><p>Пространство <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=\mathbf {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbf {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748f277e18fc2e610556425f54f51595d3fa19de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.228ex; height:2.343ex;" alt="{\displaystyle X=\mathbf {C} ^{n}}"></span> естественно вложено в спектр кольца многочленов <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 Y=\mathrm {Spec} \,\mathbf {C} [z_{1},\;z_{2},\;\ldots ,\;z_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\mathrm {Spec} \,\mathbf {C} [z_{1},\;z_{2},\;\ldots ,\;z_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/357f0837c37571b56d582a204b0e8742d0500ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.852ex; height:2.843ex;" alt="{\displaystyle Y=\mathrm {Spec} \,\mathbf {C} [z_{1},\;z_{2},\;\ldots ,\;z_{n}]}"></span> (оно совпадает с множеством всех его замкнутых точек), и топология Зарисского на <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> </semantics> </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> совпадает с той, что индуцирована топологией пространства <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Непрерывные_отображения"><span id=".D0.9D.D0.B5.D0.BF.D1.80.D0.B5.D1.80.D1.8B.D0.B2.D0.BD.D1.8B.D0.B5_.D0.BE.D1.82.D0.BE.D0.B1.D1.80.D0.B0.D0.B6.D0.B5.D0.BD.D0.B8.D1.8F"></span>Непрерывные отображения</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=9" title="Редактировать раздел «Непрерывные отображения»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=9" title="Редактировать код раздела «Непрерывные отображения»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Понятие топологии является минимально необходимым для того, чтобы говорить о <a href="/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Непрерывное отображение">непрерывных отображениях</a>. Интуитивно непрерывность есть отсутствие разрывов, то есть близкие точки при непрерывном отображении должны переходить в близкие. Оказывается, для определения понятия близости точек можно обойтись без понятия расстояния. Именно это и есть топологическое определение непрерывного отображения. </p><p>Отображение топологических пространств <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 f:(X,\;{\mathcal {T}}_{X})\to (Y,\;{\mathcal {T}}_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:(X,\;{\mathcal {T}}_{X})\to (Y,\;{\mathcal {T}}_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f75b224b1ab00770145cd434ef59128c9c7c9890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.212ex; height:2.843ex;" alt="{\displaystyle f:(X,\;{\mathcal {T}}_{X})\to (Y,\;{\mathcal {T}}_{Y})}"></span> называется <i>непрерывным</i>, если прообраз всякого открытого множества открыт. </p><p><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D1%85_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2" title="Категория топологических пространств">Категория топологических пространств</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 \mathrm {Top} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Top} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2afabdd247c34b6d2fe07c7a8f116ddcb877c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.133ex; height:2.509ex;" alt="{\displaystyle \mathrm {Top} }"></span> в качестве объектов содержит все топологические пространства, а морфизмы — непрерывные отображения. Попыткам классифицировать объекты этой категории при помощи алгебраических инвариантов посвящён раздел математической науки, который называется <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Алгебраическая топология">алгебраической топологией</a>. Изучению понятий непрерывности, а также других понятий, таких как компактность или отделимость, как таковых, без обращения к другим инструментам, посвящена <a href="/wiki/%D0%9E%D0%B1%D1%89%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Общая топология">общая топология</a>. В качестве дополнительных структур на объекте <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\in \mathrm {Ob} \,{\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in \mathrm {Ob} \,{\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245b31af953bd0afb7bd286ecf7fd3513acd4c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.244ex; height:2.343ex;" alt="{\displaystyle X\in \mathrm {Ob} \,{\mathcal {T}}}"></span> могут быть, например, пучок множеств на <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> </semantics> </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> или афинная прямая на <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> </semantics> </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>, то есть <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 \mathbb {A} _{X}\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{X}\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d29c4612520b36780bccfab3cf38db8f44f1a138" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.904ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{X}\to X}"></span>. Обозначим категорию пространств из <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 {T}}}"> <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">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> с дополнительной структурой через <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 {T}}^{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}^{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3027f8b117098a55d15e27ab2d198478f6489686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.624ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}^{E}}"></span>. <a href="/wiki/%D0%97%D0%B0%D0%B1%D1%8B%D0%B2%D0%B0%D1%8E%D1%89%D0%B8%D0%B9_%D1%84%D1%83%D0%BD%D0%BA%D1%82%D0%BE%D1%80" title="Забывающий функтор">Забывающий функтор</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 {T}}^{E}\to {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}^{E}\to {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b864bc2ea15db12b911344f6833ca843afe00c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.174ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}^{E}\to {\mathcal {T}}}"></span> — декартовы расслоения. Объекты <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 \mathrm {Ob} \,{\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ob} \,{\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08fb5605f53de848d7188bb0ffffc80182166bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.424ex; height:2.343ex;" alt="{\displaystyle \mathrm {Ob} \,{\mathcal {T}}}"></span> называются пространствами со структурой. Объект слоя <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 {T}}^{E}\to {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}^{E}\to {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b864bc2ea15db12b911344f6833ca843afe00c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.174ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}^{E}\to {\mathcal {T}}}"></span> над <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\in \mathrm {Ob} \,{\mathcal {T}}^{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in \mathrm {Ob} \,{\mathcal {T}}^{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb979fcc0d564b871441038da4fb34628161507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.933ex; height:2.676ex;" alt="{\displaystyle X\in \mathrm {Ob} \,{\mathcal {T}}^{E}}"></span> называется структурой над <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\in \mathrm {Ob} \,{\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in \mathrm {Ob} \,{\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245b31af953bd0afb7bd286ecf7fd3513acd4c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.244ex; height:2.343ex;" alt="{\displaystyle X\in \mathrm {Ob} \,{\mathcal {T}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Функциональная_структура"><span id=".D0.A4.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D0.BE.D0.BD.D0.B0.D0.BB.D1.8C.D0.BD.D0.B0.D1.8F_.D1.81.D1.82.D1.80.D1.83.D0.BA.D1.82.D1.83.D1.80.D0.B0"></span>Функциональная структура</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit&section=10" title="Редактировать раздел «Функциональная структура»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit&section=10" title="Редактировать код раздела «Функциональная структура»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>По Хохшильду функциональная структура на <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> </semantics> </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> — отображение <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 {F}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f147f366acb0ddbcd7fbe40341dcd5c9c3e509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.303ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{X}}"></span>, ставящее в соответствие каждому открытому множеству <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\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01cf5893c47ae0bfe4df06f73175c8d35bd68fa" 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}"></span> подалгебру <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 {F}}_{X}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7357c94e2866757f22a7491b5a58ad2acf86520e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.895ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{X}(U)}"></span> алгебры непрерывных вещественнозначных функций на <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}"></span>. Это отображение является <a href="/wiki/%D0%9F%D1%83%D1%87%D0%BE%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Пучок (математика)">пучком</a> алгебр — подпучком ростков непрерывных вещественнозначных функций на <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> </semantics> </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>, который содержит постоянный пучок. Это следует из условий, накладываемых на <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 {F}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f147f366acb0ddbcd7fbe40341dcd5c9c3e509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.303ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{X}}"></span>: </p> <ul><li>если <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=\bigcup _{\alpha }U_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\bigcup _{\alpha }U_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea0fba946a12f9468e924462cf658575772b4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.721ex; height:5.509ex;" alt="{\displaystyle U=\bigcup _{\alpha }U_{\alpha }}"></span> является произвольным объединением открытых множеств, то отображение <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 f:U\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f481901ff501baa824d1eab35eba6d9410ba57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.29ex; height:2.509ex;" alt="{\displaystyle f:U\to \mathbb {R} }"></span> принадлежит <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 {F}}_{X}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7357c94e2866757f22a7491b5a58ad2acf86520e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.895ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{X}(U)}"></span> в том случае, когда ограничение <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> на каждое открытое множество <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_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c507812c8cdaf4cea8d2e7e1705b495a3010a352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.872ex; height:2.509ex;" alt="{\displaystyle U_{\alpha }}"></span> принадлежит <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 {F}}_{X}(U_{\alpha })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}(U_{\alpha })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba03fa6d3e720b34ffa6dea64c467ebdc399ea1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.984ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{X}(U_{\alpha })}"></span>;</li> <li><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 {F}}_{X}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7357c94e2866757f22a7491b5a58ad2acf86520e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.895ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{X}(U)}"></span> содержит все постоянные на <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}"></span> функции;</li> <li>если <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\subset U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊂</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subset U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae37b0495b78ebcc4b8df66949ebd5659a4fafb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle V\subset U}"></span>, то ограничение <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 f\in {\mathcal {F}}_{X}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {F}}_{X}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84c5923494b28005a752d2a89784e517d07a7efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.015ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {F}}_{X}(U)}"></span> на <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span> содержится в <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 {F}}_{X}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{X}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7c23ff8fe2f963341db1346b40579bbb0ed970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.9ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{X}(V)}"></span>.</li></ul> <p>Например, <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 C^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971ed05871d69309df32efdfd2020128c9cf69d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}"></span>-многообразие с краем является паракомпактным хаусдорфовым пространством, наделенным функциональной структурой, <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 (M,{\mathcal {F}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,{\mathcal {F}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3757c0440337918f29bcc093aecfcf0da6db8464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.212ex; height:2.843ex;" alt="{\displaystyle (M,{\mathcal {F}})}"></span>, локально изоморфным пространству <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 (\mathbb {R} _{+}^{n},C^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>,</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} _{+}^{n},C^{\infty })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db3a2375aaf3aa15f5cbd8c0b19075cdd1b954d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.705ex; height:3.009ex;" alt="{\displaystyle (\mathbb {R} _{+}^{n},C^{\infty })}"></span>. Край состоит из тех точек, которые переводятся картами в точки гиперплоскости, являясь гладким <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 (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}"></span>-мерным многообразием с индуцированной структурой. </p> <div class="mw-heading mw-heading2"><h2 id="См._также"><span 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a::after{content:"";width:1.1em;height:1.1em;display:inline-block;vertical-align:middle;background-position:center;background-repeat:no-repeat;background-size:contain}.mw-parser-output .id-lock-free.id-lock-free a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")}.mw-parser-output .id-lock-limited.id-lock-limited a::after,.mw-parser-output .id-lock-registration.id-lock-registration a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")}.mw-parser-output .id-lock-subscription.id-lock-subscription a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")}.mw-parser-output .cs1-ws-icon a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")}.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}html.skin-theme-clientpref-night .mw-parser-output .id-lock-free a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-limited a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-registration a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-subscription a::after{filter:invert(1)hue-rotate(180deg)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-os .mw-parser-output .id-lock-free a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-limited a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-registration a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-subscription a::after{filter:invert(1)hue-rotate(180deg)}}</style><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Александров, П. С.</i> Введение в теорию множеств и общую топологию. — <abbr title="Москва">М.</abbr>: ГИИТЛ, 1948.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Келли, Дж. Л.</i> Общая топология. — <abbr title="Москва">М.</abbr>: Наука, 1968.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Энгелькинг, Р.</i> Общая топология. — <abbr title="Москва">М.</abbr>: <a href="/wiki/%D0%9C%D0%B8%D1%80_(%D0%B8%D0%B7%D0%B4%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE)" title="Мир (издательство)">Мир</a>, 1986. — 752 с.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFФрёлихер1970"><i>Фрёлихер, А., Бухер В.</i> Дифференциальное исчисление в векторных пространствах без нормы. — <abbr title="Москва">М.</abbr>: Мир, 1970.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Виро О. Я., Иванов О. А., Нецветаев Н. Ю., Харламов В. М.</i> Элементарная топология.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Васильев В. А.</i> Введение в топологию. — <abbr title="Москва">М.</abbr>: Фазис, 1997.</span></li></ul> <div role="navigation" class="navbox" aria-labelledby="Топология" data-name="Топология"><table class="nowraplinks collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="colgroup" class="navbox-title" colspan="2"><span class="navbox-gear" style="float:left;text-align:left;width:5em;margin-right:0.5em"><span class="noprint skin-invert-image" typeof="mw:File"><a href="/wiki/%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Перейти к шаблону «Топология»"><img alt="Перейти к шаблону «Топология»" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/14px-Wikipedia_interwiki_section_gear_icon.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/21px-Wikipedia_interwiki_section_gear_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/28px-Wikipedia_interwiki_section_gear_icon.svg.png 2x" data-file-width="14" data-file-height="14" /></a></span></span><div id="Топология" style="font-size:114%;margin:0 5em"><a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Топология">Топология</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1px"><a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F#Разделы_топологии" title="Топология">Разделы</a></th><td class="navbox-list navbox-odd hlist hlist-items-nowrap" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%D0%9E%D0%B1%D1%89%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Общая топология">Теоретико-множественная</a></li> <li><a href="/wiki/%D0%9C%D0%B0%D0%BB%D0%BE%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Маломерная топология">Маломерная</a> <ul><li><a href="/wiki/%D0%94%D0%B2%D1%83%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" class="mw-redirect" title="Двумерная топология">двумерная</a></li> <li><a href="/wiki/%D0%A2%D1%80%D1%91%D1%85%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Трёхмерная топология">трёхмерная</a></li> <li><a href="/wiki/%D0%A7%D0%B5%D1%82%D1%8B%D1%80%D1%91%D1%85%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Четырёхмерная топология">четырёхмерная</a></li></ul></li> <li><a href="/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" class="mw-redirect" title="Дифференциальная топология">Дифференциальная</a></li> <li><span data-interwiki-lang="en" data-interwiki-article="Geometric topology"><a href="/w/index.php?title=%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Геометрическая топология (страница отсутствует)">Геометрическая</a></span><sup class="noprint" style="font-style:normal; font-weight:normal;"><a href="https://en.wikipedia.org/wiki/Geometric_topology" class="extiw" title="en:Geometric topology"><span title="Geometric topology — версия статьи «Геометрическая топология» на английском языке">[англ.]</span></a></sup> <ul><li><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%83%D0%B7%D0%BB%D0%BE%D0%B2" title="Теория узлов">теория узлов</a></li> <li><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BA%D0%BE%D1%81" title="Теория кос">теория кос</a></li></ul></li> <li><a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" class="mw-redirect" title="Комбинаторная топология">Комбинаторная</a></li> <li><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B3%D0%BE%D0%BC%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D0%B9" class="mw-redirect" title="Теория гомотопий">Гомотопическая</a></li> <li><a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Алгебраическая топология">Алгебраическая</a></li> <li><a href="/wiki/%D0%92%D1%8B%D1%87%D0%B8%D1%81%D0%BB%D0%B8%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Вычислительная топология">Алгоритмическая</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1px"><a href="/wiki/%D0%93%D0%BB%D0%BE%D1%81%D1%81%D0%B0%D1%80%D0%B8%D0%B9_%D0%BE%D0%B1%D1%89%D0%B5%D0%B9_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B8" title="Глоссарий общей топологии">Основные понятия</a></th><td class="navbox-list navbox-even hlist hlist-items-nowrap" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрическое пространство">Метрическое пространство</a></li> <li><a class="mw-selflink selflink">Топологическое пространство</a></li> <li><a href="/wiki/%D0%93%D0%BE%D0%BC%D0%B5%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Гомеоморфизм">Гомеоморфизм</a></li> <li><a href="/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B8%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплициальный комплекс">Симплициальный комплекс</a></li> <li><a href="/wiki/CW-%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="CW-комплекс">CW-комплекс</a></li> <li><a href="/wiki/%D0%92%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Вложение">Вложение</a></li> <li><a href="/wiki/%D0%9F%D1%83%D1%82%D1%8C_(%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F)" title="Путь (топология)">Путь</a> <ul><li><a href="/wiki/%D0%9F%D0%B5%D1%82%D0%BB%D1%8F_(%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F)" title="Петля (топология)">петля</a></li></ul></li> <li><a href="/wiki/%D0%93%D0%BE%D0%BC%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D1%8F" title="Гомотопия">Гомотопия</a> <ul><li><a href="/wiki/%D0%A0%D0%B5%D1%82%D1%80%D0%B0%D0%BA%D1%82" title="Ретракт">ретракция</a></li> <li><a href="/wiki/%D0%98%D0%B7%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D1%8F" title="Изотопия">изотопия</a></li> <li><a href="/wiki/%D0%9E%D0%B1%D1%8A%D0%B5%D0%BC%D0%BB%D1%8E%D1%89%D0%B0%D1%8F_%D0%B8%D0%B7%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D1%8F" title="Объемлющая изотопия">объемлющая изотопия</a></li></ul></li> <li><a href="/wiki/%D0%A0%D0%B0%D1%81%D1%81%D0%BB%D0%BE%D0%B5%D0%BD%D0%B8%D0%B5" title="Расслоение">Расслоение</a></li> <li><a href="/wiki/%D0%9F%D1%83%D1%87%D0%BE%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Пучок (математика)">Пучок</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1px"><a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Многообразие">Многообразия</a></th><td class="navbox-list navbox-odd hlist hlist-items-nowrap" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Поверхность">Поверхность</a></li> <li><a href="/wiki/%D0%A2%D1%80%D1%91%D1%85%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Трёхмерное многообразие">Трёхмерное многообразие</a></li> <li><a href="/wiki/%D0%9E%D1%80%D0%B8%D0%B5%D0%BD%D1%82%D0%B0%D1%86%D0%B8%D1%8F" title="Ориентация">Ориентация</a></li> <li><a href="/wiki/%D0%A1%D0%B2%D1%8F%D0%B7%D0%BD%D0%B0%D1%8F_%D1%81%D1%83%D0%BC%D0%BC%D0%B0" title="Связная сумма">Связная сумма</a></li> <li><a href="/wiki/%D0%A0%D0%B0%D0%B7%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D1%80%D1%83%D1%87%D0%BA%D0%B8" title="Разложение на ручки">Разложение на ручки</a></li> <li><a href="/wiki/%D0%A1%D0%BB%D0%BE%D0%B5%D0%BD%D0%B8%D0%B5" title="Слоение">Слоение</a></li> <li><a href="/wiki/%D0%91%D0%BE%D1%80%D0%B4%D0%B8%D0%B7%D0%BC" title="Бордизм">Бордизм</a></li> <li><a href="/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B7%D0%B0%D1%86%D0%B5%D0%BF%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Коэффициент зацепления">Коэффициент зацепления</a></li> <li><a href="/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8C_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D1%8F" title="Степень отображения">Степень отображения</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1px"><a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B8%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82" class="mw-redirect" title="Топологический инвариант">Топологические инварианты</a></th><td class="navbox-list navbox-even hlist hlist-items-nowrap" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%D0%A1%D0%B2%D1%8F%D0%B7%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Связное пространство">Связность</a></li> <li><a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Компактное пространство">Компактность</a></li> <li><a href="/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%85%D0%B0%D1%80%D0%B0%D0%BA%D1%82%D0%B5%D1%80%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0" title="Эйлерова характеристика">Эйлерова характеристика</a></li> <li><a href="/wiki/%D0%9F%D0%B5%D1%80%D0%B2%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B9" title="Первая группа гомологий">Первая группа гомологий</a></li> <li><a href="/wiki/%D0%9F%D0%B5%D1%80%D0%B2%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%BA%D0%BE%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B9" title="Первая группа когомологий">Первая группа когомологий</a></li> <li><a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%91%D0%B5%D1%82%D1%82%D0%B8" title="Число Бетти">Числа Бетти</a></li> <li><a href="/wiki/%D0%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%B0" title="Размерность пространства">Размерность</a></li> <li><a href="/wiki/%D0%A4%D1%83%D0%BD%D0%B4%D0%B0%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Фундаментальная группа">Фундаментальная группа</a></li> <li><a href="/wiki/%D0%9A%D0%BE%D0%BD%D1%84%D0%B8%D0%B3%D1%83%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE_(%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F)" title="Конфигурационное пространство (топология)">Конфигурационное пространство</a></li> <li><a href="/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%BA%D0%BB%D0%B0%D1%81%D1%81%D0%BE%D0%B2_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B9_%D0%BF%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Группа классов преобразований поверхности">Группа классов отображений</a></li> <li><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B9" title="Теория гомологий">Когомологии</a></li> <li><a href="/wiki/%D0%93%D0%BE%D0%BC%D0%BE%D1%82%D0%BE%D0%BF%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D1%8B" title="Гомотопические группы">Гомотопические группы</a></li></ul> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r137874932">.mw-parser-output .ambox{border:1px solid var(--border-color-base,#a2a9b1);border-left:10px solid #36c;background:var(--background-color-neutral-subtle,#f8f9fa);box-sizing:border-box;margin:0 10%}html 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class="plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9A%D0%B0%D0%BA_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D1%8C_%D1%81%D1%82%D0%B0%D1%82%D1%8C%D0%B8" title="Улучшение статьи"><img alt="Улучшение статьи" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-div"><b>Для улучшения этой статьи по математике <a href="/wiki/%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:Rq/%D0%9F%D0%BE%D1%8F%D1%81%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F" title="Шаблон:Rq/Пояснения">желательно</a>:</b><ul style="margin-top: 0;"> <li>Проставить <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BD%D0%BE%D1%81%D0%BA%D0%B8" title="Википедия:Сноски">сноски</a>, внести более точные указания на источники.</li><li>Добавить <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%98%D0%BB%D0%BB%D1%8E%D1%81%D1%82%D1%80%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5" title="Википедия:Иллюстрирование">иллюстрации</a>.</li></ul></div><div class="mbox-textsmall-div hide-when-compact">После исправления проблемы исключите её из списка. Удалите шаблон, если устранены все недостатки.</div></td></tr></tbody></table> <div role="navigation" class="navbox" aria-labelledby="Ссылки_на_внешние_ресурсы" data-name="External links" style="padding-top:1px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="colgroup" class="navbox-title" colspan="2" style="display:none"><span class="navbox-gear" style="float:left;text-align:left;width:5em;margin-right:0.5em"><span class="noprint skin-invert-image" typeof="mw:File"><a href="/wiki/%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:External_links" title="Перейти к шаблону «External links»"><img alt="Перейти к шаблону «External links»" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/14px-Wikipedia_interwiki_section_gear_icon.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/21px-Wikipedia_interwiki_section_gear_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/28px-Wikipedia_interwiki_section_gear_icon.svg.png 2x" data-file-width="14" data-file-height="14" /></a></span></span><div id="Ссылки_на_внешние_ресурсы" style="font-size:114%;margin:0 5em">Ссылки на внешние ресурсы</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1px"><div style="padding: 0 35px 0 0; width: 100%;"><div class="skin-invert-image" style="float: left;"><span class="noprint skin-invert-image" typeof="mw:File"><a href="/wiki/%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:%D0%92%D0%BD%D0%B5%D1%88%D0%BD%D0%B8%D0%B5_%D1%81%D1%81%D1%8B%D0%BB%D0%BA%D0%B8" title="Перейти к шаблону «Внешние ссылки»"><img alt="Перейти к шаблону «Внешние ссылки»" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/14px-Wikipedia_interwiki_section_gear_icon.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/21px-Wikipedia_interwiki_section_gear_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Wikipedia_interwiki_section_gear_icon.svg/28px-Wikipedia_interwiki_section_gear_icon.svg.png 2x" data-file-width="14" data-file-height="14" /></a></span> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q179899#identifiers" title="Перейти к элементу Викиданных"><img alt="Перейти к элементу Викиданных" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/14px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/21px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/28px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div>  Тематические сайты</div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TopologicalSpace.html">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/topological%20space">nLab</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1px">Словари и энциклопедии</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=8313&url_prefix=https://denstoredanske.lex.dk/&id=topologisk_rum">Большая датская</a></li> <li><a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/ec-gec-0229985.xml">Большая каталанская</a></li> <li><a rel="nofollow" class="external text" href="https://bigenc.ru/c/topologicheskoe-prostranstvo-e185d9">Большая российская (научно-образовательный портал)</a></li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/topological-space">Britannica (онлайн)</a></li> <li><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3988109.html">PWN</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1px">В библиографических каталогах</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%D0%9D%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B1%D0%B8%D0%B1%D0%BB%D0%B8%D0%BE%D1%82%D0%B5%D0%BA%D0%B0_%D0%A4%D1%80%D0%B0%D0%BD%D1%86%D0%B8%D0%B8" title="Национальная библиотека Франции">BNF</a>: <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb13162791j">13162791j</a></li> <li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4137586-5">4137586-5</a></li> <li><a href="/wiki/%D0%9D%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B1%D0%B8%D0%B1%D0%BB%D0%B8%D0%BE%D1%82%D0%B5%D0%BA%D0%B0_%D0%98%D0%B7%D1%80%D0%B0%D0%B8%D0%BB%D1%8F" title="Национальная библиотека Израиля">J9U</a>: <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007541466005171">987007541466005171</a></li> <li><a href="/wiki/%D0%9A%D0%BE%D0%BD%D1%82%D1%80%D0%BE%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BD%D0%BE%D0%BC%D0%B5%D1%80_%D0%91%D0%B8%D0%B1%D0%BB%D0%B8%D0%BE%D1%82%D0%B5%D0%BA%D0%B8_%D0%9A%D0%BE%D0%BD%D0%B3%D1%80%D0%B5%D1%81%D1%81%D0%B0" title="Контрольный номер Библиотеки Конгресса">LCCN</a>: <a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85136087">sh85136087</a></li> <li><a href="/wiki/%D0%9D%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D0%B0%D1%80%D0%BB%D0%B0%D0%BC%D0%B5%D0%BD%D1%82%D1%81%D0%BA%D0%B0%D1%8F_%D0%B1%D0%B8%D0%B1%D0%BB%D0%B8%D0%BE%D1%82%D0%B5%D0%BA%D0%B0_(%D0%AF%D0%BF%D0%BE%D0%BD%D0%B8%D1%8F)" title="Национальная парламентская библиотека (Япония)">NDL</a>: <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00564270">00564270</a></li></ul> </div></td></tr></tbody></table></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Источник — <a dir="ltr" href="https://ru.wikipedia.org/w/index.php?title=Топологическое_пространство&oldid=139947866">https://ru.wikipedia.org/w/index.php?title=Топологическое_пространство&oldid=139947866</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8" title="Служебная:Категории">Категории</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Категория:Топология">Топология</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%B0" title="Категория:Топологические пространства">Топологические пространства</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%9E%D0%B1%D1%89%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Категория:Общая топология">Общая топология</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A2%D0%B8%D0%BF%D1%8B_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D1%85_%D0%BE%D0%B1%D1%8A%D0%B5%D0%BA%D1%82%D0%BE%D0%B2" title="Категория:Типы математических объектов">Типы математических объектов</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Скрытые категории: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D0%B1%D0%B5%D0%B7_%D1%81%D0%BD%D0%BE%D1%81%D0%BE%D0%BA" title="Категория:Википедия:Статьи без сносок">Википедия:Статьи без сносок</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D0%B1%D0%B5%D0%B7_%D0%B8%D0%B7%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B9_(%D1%83%D0%BA%D0%B0%D0%B7%D0%B0%D0%BD%D0%BE_%D0%B2_%D0%92%D0%B8%D0%BA%D0%B8%D0%B4%D0%B0%D0%BD%D0%BD%D1%8B%D1%85:_P18)" title="Категория:Википедия:Статьи без изображений (указано в Викиданных: P18)">Википедия:Статьи без изображений (указано в Викиданных: P18)</a></li><li><a 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</div> </div> <div id="mw-navigation"> <h2>Навигация</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Персональные инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Страница участника для моего IP">Вы не представились системе</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D1%91_%D0%BE%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Страница обсуждений для моего IP [n]" accesskey="n"><span>Обсуждение</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D0%B9_%D0%B2%D0%BA%D0%BB%D0%B0%D0%B4" title="Список правок, сделанных с этого IP-адреса [y]" accesskey="y"><span>Вклад</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BE%D0%B7%D0%B4%D0%B0%D1%82%D1%8C_%D1%83%D1%87%D1%91%D1%82%D0%BD%D1%83%D1%8E_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81%D1%8C&returnto=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5+%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Мы предлагаем вам создать учётную запись и войти в систему, хотя это и не обязательно."><span>Создать учётную запись</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%92%D1%85%D0%BE%D0%B4&returnto=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5+%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Здесь можно зарегистрироваться в системе, но это необязательно. [o]" accesskey="o"><span>Войти</span></a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Пространства имён</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Просмотреть контентную страницу [c]" accesskey="c"><span>Статья</span></a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/%D0%9E%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5:%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" rel="discussion" title="Обсуждение основной страницы [t]" accesskey="t"><span>Обсуждение</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">русский</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Просмотры</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected mw-list-item"><a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE"><span>Читать</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&veaction=edit" title="Редактировать данную страницу [v]" accesskey="v"><span>Править</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=edit" title="Править исходный текст этой страницы [e]" accesskey="e"><span>Править код</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE&action=history" title="Журнал изменений страницы [h]" accesskey="h"><span>История</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Больше возможностей" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Ещё</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Поиск</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Искать в Википедии" aria-label="Искать в Википедии" autocapitalize="sentences" title="Искать в Википедии [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Служебная:Поиск"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Найти страницы, содержащие указанный текст" value="Найти"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Перейти к странице, имеющей в точности такое название" value="Перейти"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Навигация</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу [z]" accesskey="z"><span>Заглавная страница</span></a></li><li id="n-content" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5"><span>Содержание</span></a></li><li id="n-featured" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%98%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D1%81%D1%82%D0%B0%D1%82%D1%8C%D0%B8" title="Статьи, считающиеся лучшими статьями проекта"><span>Избранные статьи</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Посмотреть случайно выбранную страницу [x]" accesskey="x"><span>Случайная статья</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB:%D0%A2%D0%B5%D0%BA%D1%83%D1%89%D0%B8%D0%B5_%D1%81%D0%BE%D0%B1%D1%8B%D1%82%D0%B8%D1%8F" title="Статьи о текущих событиях в мире"><span>Текущие события</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ru.wikipedia.org&uselang=ru" title="Поддержите нас"><span>Пожертвовать</span></a></li> </ul> </div> </nav> <nav id="p-participation" class="mw-portlet mw-portlet-participation vector-menu-portal portal vector-menu" aria-labelledby="p-participation-label" > <h3 id="p-participation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Участие</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-bug_in_article" class="mw-list-item"><a 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class="vector-menu-heading-label">В других проектах</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Topology" hreflang="en"><span>Викисклад</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q179899" title="Ссылка на связанный элемент репозитория данных [g]" accesskey="g"><span>Элемент Викиданных</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">На других языках</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D8%B7%D9%88%D8%A8%D9%88%D9%84%D9%88%D8%AC%D9%8A" title="فضاء طوبولوجي — арабский" lang="ar" hreflang="ar" data-title="فضاء طوبولوجي" data-language-autonym="العربية" data-language-local-name="арабский" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_topol%C3%B3xicu" title="Espaciu topolóxicu — астурийский" lang="ast" hreflang="ast" data-title="Espaciu topolóxicu" data-language-autonym="Asturianu" data-language-local-name="астурийский" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BF%D0%B0%D0%BB%D0%B0%D0%B3%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B0%D1%81%D1%82%D0%BE%D1%80%D0%B0" title="Тапалагічная прастора — белорусский" lang="be" hreflang="be" data-title="Тапалагічная прастора" data-language-autonym="Беларуская" data-language-local-name="белорусский" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Топологично пространство — болгарский" lang="bg" hreflang="bg" data-title="Топологично пространство" data-language-autonym="Български" data-language-local-name="болгарский" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_topol%C3%B2gic" title="Espai topològic — каталанский" lang="ca" hreflang="ca" data-title="Espai topològic" data-language-autonym="Català" data-language-local-name="каталанский" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Topologick%C3%BD_prostor" title="Topologický prostor — чешский" lang="cs" hreflang="cs" data-title="Topologický prostor" data-language-autonym="Čeština" data-language-local-name="чешский" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%BB%D0%BB%C4%95_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Топологиллĕ уçлăх — чувашский" lang="cv" hreflang="cv" data-title="Топологиллĕ уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="чувашский" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_topolegol" title="Gofod topolegol — валлийский" lang="cy" hreflang="cy" data-title="Gofod topolegol" data-language-autonym="Cymraeg" data-language-local-name="валлийский" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Topologisk_rum" title="Topologisk rum — датский" lang="da" hreflang="da" data-title="Topologisk rum" data-language-autonym="Dansk" data-language-local-name="датский" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Topologischer_Raum" title="Topologischer Raum — немецкий" lang="de" hreflang="de" data-title="Topologischer Raum" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%BF%CF%80%CE%BF%CE%BB%CE%BF%CE%B3%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Τοπολογικός χώρος — греческий" lang="el" hreflang="el" data-title="Τοπολογικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="греческий" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Topological_space" title="Topological space — английский" lang="en" hreflang="en" data-title="Topological space" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Topologia_spaco" title="Topologia spaco — эсперанто" lang="eo" hreflang="eo" data-title="Topologia spaco" data-language-autonym="Esperanto" data-language-local-name="эсперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_topol%C3%B3gico" title="Espacio topológico — испанский" lang="es" hreflang="es" data-title="Espacio topológico" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Topoloogiline_ruum" title="Topoloogiline ruum — эстонский" lang="et" hreflang="et" data-title="Topoloogiline ruum" data-language-autonym="Eesti" data-language-local-name="эстонский" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Espazio_topologiko" title="Espazio topologiko — баскский" lang="eu" hreflang="eu" data-title="Espazio topologiko" data-language-autonym="Euskara" data-language-local-name="баскский" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D8%AA%D9%88%D9%BE%D9%88%D9%84%D9%88%DA%98%DB%8C" title="فضای توپولوژی — персидский" lang="fa" hreflang="fa" data-title="فضای توپولوژی" data-language-autonym="فارسی" data-language-local-name="персидский" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Topologinen_avaruus" title="Topologinen avaruus — финский" lang="fi" hreflang="fi" data-title="Topologinen avaruus" data-language-autonym="Suomi" data-language-local-name="финский" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_topologique" title="Espace topologique — французский" lang="fr" hreflang="fr" data-title="Espace topologique" data-language-autonym="Français" data-language-local-name="французский" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_topol%C3%B3xico" title="Espazo topolóxico — галисийский" lang="gl" hreflang="gl" data-title="Espazo topolóxico" data-language-autonym="Galego" data-language-local-name="галисийский" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99" title="מרחב טופולוגי — иврит" lang="he" hreflang="he" data-title="מרחב טופולוגי" data-language-autonym="עברית" data-language-local-name="иврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Topolo%C5%A1ki_prostor" title="Topološki prostor — хорватский" lang="hr" hreflang="hr" data-title="Topološki prostor" data-language-autonym="Hrvatski" data-language-local-name="хорватский" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Topologikus_t%C3%A9r" title="Topologikus tér — венгерский" lang="hu" hreflang="hu" data-title="Topologikus tér" data-language-autonym="Magyar" data-language-local-name="венгерский" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_topologis" title="Ruang topologis — индонезийский" lang="id" hreflang="id" data-title="Ruang topologis" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезийский" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_topologico" title="Spazio topologico — итальянский" lang="it" hreflang="it" data-title="Spazio topologico" data-language-autonym="Italiano" data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BD%8D%E7%9B%B8%E7%A9%BA%E9%96%93" title="位相空間 — японский" lang="ja" hreflang="ja" data-title="位相空間" data-language-autonym="日本語" data-language-local-name="японский" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%9D%E1%83%9E%E1%83%9D%E1%83%9A%E1%83%9D%E1%83%92%E1%83%98%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A1%E1%83%98%E1%83%95%E1%83%A0%E1%83%AA%E1%83%94" title="ტოპოლოგიური სივრცე — грузинский" lang="ka" hreflang="ka" data-title="ტოპოლოგიური სივრცე" data-language-autonym="ქართული" data-language-local-name="грузинский" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D0%BA%D0%B5%D2%A3%D1%96%D1%81%D1%82%D1%96%D0%BA" title="Топологиялық кеңістік — казахский" lang="kk" hreflang="kk" data-title="Топологиялық кеңістік" data-language-autonym="Қазақша" data-language-local-name="казахский" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%84%EC%83%81_%EA%B3%B5%EA%B0%84_(%EC%88%98%ED%95%99)" title="위상 공간 (수학) — корейский" lang="ko" hreflang="ko" data-title="위상 공간 (수학)" data-language-autonym="한국어" data-language-local-name="корейский" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Topolo%C4%A3iska_telpa" title="Topoloģiska telpa — латышский" lang="lv" hreflang="lv" data-title="Topoloģiska telpa" data-language-autonym="Latviešu" data-language-local-name="латышский" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_topologi" title="Ruang topologi — малайский" lang="ms" hreflang="ms" data-title="Ruang topologi" data-language-autonym="Bahasa Melayu" data-language-local-name="малайский" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Topologische_ruimte" title="Topologische ruimte — нидерландский" lang="nl" hreflang="nl" data-title="Topologische ruimte" data-language-autonym="Nederlands" data-language-local-name="нидерландский" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Topologisk_rom" title="Topologisk rom — нюнорск" lang="nn" hreflang="nn" data-title="Topologisk rom" data-language-autonym="Norsk nynorsk" data-language-local-name="нюнорск" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9F%E0%A9%8C%E0%A8%AA%E0%A9%8C%E0%A8%B2%E0%A9%8C%E0%A8%9C%E0%A9%80%E0%A8%95%E0%A8%B2_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਟੌਪੌਲੌਜੀਕਲ ਸਪੇਸ — панджаби" lang="pa" hreflang="pa" data-title="ਟੌਪੌਲੌਜੀਕਲ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="панджаби" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_topologiczna" title="Przestrzeń topologiczna — польский" lang="pl" hreflang="pl" data-title="Przestrzeń topologiczna" data-language-autonym="Polski" data-language-local-name="польский" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_topol%C3%B2gich" title="Spassi topològich — Piedmontese" lang="pms" hreflang="pms" data-title="Spassi topològich" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%B9%D9%88%D9%BE%D9%88%D9%84%D9%88%D8%AC%DB%8C%DA%A9%D9%84_%D8%B3%D9%BE%DB%8C%D8%B3" title="ٹوپولوجیکل سپیس — Western Punjabi" lang="pnb" hreflang="pnb" data-title="ٹوپولوجیکل سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_topol%C3%B3gico" title="Espaço topológico — португальский" lang="pt" hreflang="pt" data-title="Espaço topológico" data-language-autonym="Português" data-language-local-name="португальский" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_topologic" title="Spațiu topologic — румынский" lang="ro" hreflang="ro" data-title="Spațiu topologic" data-language-autonym="Română" data-language-local-name="румынский" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Topological_space" title="Topological space — Simple English" lang="en-simple" hreflang="en-simple" data-title="Topological space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Topologick%C3%BD_priestor" title="Topologický priestor — словацкий" lang="sk" hreflang="sk" data-title="Topologický priestor" data-language-autonym="Slovenčina" data-language-local-name="словацкий" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Topolo%C5%A1ki_prostor" title="Topološki prostor — словенский" lang="sl" hreflang="sl" data-title="Topološki prostor" data-language-autonym="Slovenščina" data-language-local-name="словенский" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_topologjike" title="Hapësira topologjike — албанский" lang="sq" hreflang="sq" data-title="Hapësira topologjike" data-language-autonym="Shqip" data-language-local-name="албанский" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D1%88%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Тополошки простор — сербский" lang="sr" hreflang="sr" data-title="Тополошки простор" data-language-autonym="Српски / srpski" data-language-local-name="сербский" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Topolojik_uzaylar" title="Topolojik uzaylar — турецкий" lang="tr" hreflang="tr" data-title="Topolojik uzaylar" data-language-autonym="Türkçe" data-language-local-name="турецкий" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Топологічний простір — украинский" lang="uk" hreflang="uk" data-title="Топологічний простір" data-language-autonym="Українська" data-language-local-name="украинский" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_t%C3%B4_p%C3%B4" title="Không gian tô pô — вьетнамский" lang="vi" hreflang="vi" data-title="Không gian tô pô" data-language-autonym="Tiếng Việt" data-language-local-name="вьетнамский" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D2%97%D0%B8%D0%BA_%D1%85%D0%B0%D1%80%D0%B0%D0%BD" title="Тополоҗик харан — калмыцкий" lang="xal" hreflang="xal" data-title="Тополоҗик харан" data-language-autonym="Хальмг" data-language-local-name="калмыцкий" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8B%93%E6%89%91%E7%A9%BA%E9%97%B4" title="拓扑空间 — китайский" lang="zh" hreflang="zh" data-title="拓扑空间" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%8B%93%E6%92%B2%E7%A9%BA%E9%96%93" title="拓撲空間 — Literary Chinese" lang="lzh" hreflang="lzh" data-title="拓撲空間" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%8B%93%E6%92%B2%E7%A9%BA%E9%96%93" title="拓撲空間 — кантонский" lang="yue" hreflang="yue" data-title="拓撲空間" data-language-autonym="粵語" data-language-local-name="кантонский" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q179899#sitelinks-wikipedia" title="Править ссылки на другие языки" class="wbc-editpage">Править ссылки</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Эта страница в последний раз была отредактирована 2 сентября 2024 в 18:25.</li> <li id="footer-info-copyright">Текст доступен по <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.ru">лицензии Creative Commons «С указанием авторства — С сохранением условий» (CC BY-SA)</a>; в отдельных случаях могут действовать дополнительные условия. <span class="noprint">Подробнее см. <a 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