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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Topologische groep</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ga naar een artikel in een andere taal. 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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%B3%D1%80%D1%83%D0%BF%D0%B0" title="Тапалагічная група – Belarussisch" lang="be" hreflang="be" data-title="Тапалагічная група" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" 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%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Топологична група – Bulgaars" lang="bg" hreflang="bg" data-title="Топологична група" data-language-autonym="Български" data-language-local-name="Bulgaars" 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/Grup_topol%C3%B2gic" title="Grup topològic – Catalaans" lang="ca" hreflang="ca" data-title="Grup topològic" data-language-autonym="Català" data-language-local-name="Catalaans" 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%A1_grupa" title="Topologická grupa – Tsjechisch" lang="cs" hreflang="cs" data-title="Topologická grupa" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Topologische_Gruppe" title="Topologische Gruppe – Duits" lang="de" hreflang="de" data-title="Topologische Gruppe" data-language-autonym="Deutsch" data-language-local-name="Duits" 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%CE%AE_%CE%BF%CE%BC%CE%AC%CE%B4%CE%B1" title="Τοπολογική ομάδα – Grieks" lang="el" hreflang="el" data-title="Τοπολογική ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" 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_group" title="Topological group – Engels" lang="en" hreflang="en" data-title="Topological group" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_topol%C3%B3gico" title="Grupo topológico – Spaans" lang="es" hreflang="es" data-title="Grupo topológico" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_%D8%AA%D9%88%D9%BE%D9%88%D9%84%D9%88%DA%98%DB%8C%DA%A9%DB%8C" title="گروه توپولوژیکی – Perzisch" lang="fa" hreflang="fa" data-title="گروه توپولوژیکی" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_topologique" title="Groupe topologique – Frans" lang="fr" hreflang="fr" data-title="Groupe topologique" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%AA" title="חבורה טופולוגית – Hebreeuws" lang="he" hreflang="he" data-title="חבורה טופולוגית" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%A8%D5%B6%D5%A4%D5%B0%D5%A1%D5%BF_%D5%AD%D5%B8%D6%82%D5%B4%D5%A2" title="Անընդհատ խումբ – Armeens" lang="hy" hreflang="hy" data-title="Անընդհատ խումբ" data-language-autonym="Հայերեն" data-language-local-name="Armeens" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_topologi" title="Grup topologi – Indonesisch" lang="id" hreflang="id" data-title="Grup topologi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" 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/Gruppo_topologico" title="Gruppo topologico – Italiaans" lang="it" hreflang="it" data-title="Gruppo topologico" data-language-autonym="Italiano" data-language-local-name="Italiaans" 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%BE%A4" title="位相群 – Japans" lang="ja" hreflang="ja" data-title="位相群" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</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%B5%B0" title="위상군 – Koreaans" lang="ko" hreflang="ko" data-title="위상군" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_topologiczna" title="Grupa topologiczna – Pools" lang="pl" hreflang="pl" data-title="Grupa topologiczna" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_topol%C3%B3gico" title="Grupo topológico – Portugees" lang="pt" hreflang="pt" data-title="Grupo topológico" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/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%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Топологическая группа – Russisch" lang="ru" hreflang="ru" data-title="Топологическая группа" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Topologisk_grupp" title="Topologisk grupp – Zweeds" lang="sv" hreflang="sv" data-title="Topologisk grupp" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</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%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Топологічна група – Oekraïens" lang="uk" hreflang="uk" data-title="Топологічна група" data-language-autonym="Українська" data-language-local-name="Oekraïens" class="interlanguage-link-target"><span>Українська</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%BE%A4" title="拓扑群 – 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encyclopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nl" dir="ltr"><p>In de <a href="/wiki/Wiskunde" title="Wiskunde">wiskunde</a> zijn de <b>topologische groepen</b> tegelijkertijd <a href="/wiki/Groep_(wiskunde)" title="Groep (wiskunde)">groepen</a> en <a href="/wiki/Topologische_ruimte" title="Topologische ruimte">topologische ruimten</a> zodanig dat de groepsstructuur en de topologische structuur compatibel zijn. Concreet betekent dit voor een groep <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 (G,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54a87abf331634c8962ef14c4c5ec41f94fd29c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}"></span> dat de vermenigvuldiging en de inversie <a href="/wiki/Continue_functie_(analyse)" title="Continue functie (analyse)">continu</a> zijn. </p><p>In deze definitie wordt de vermenigvuldiging opgevat als een afbeelding van het <a href="/wiki/Cartesisch_product" title="Cartesisch product">Cartesisch product</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 G\times G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>×</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\times G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47122fd4c5df97c4074ffcab5774f0a46018668f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.494ex; height:2.176ex;" alt="{\displaystyle G\times G}"></span>, uitgerust met de <a href="/wiki/Producttopologie" title="Producttopologie">producttopologie</a>, naar <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> zelf. </p><p>Veel auteurs eisen dat <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> als topologische ruimte een <a href="/wiki/Hausdorff-ruimte" title="Hausdorff-ruimte">Hausdorff-ruimte</a> is. </p><p>De continuïteit van zowel vermenigvuldiging als inversie kan worden samengevat in de eis dat de afbeelding </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 G\times G\to G:(x,y)\mapsto xy^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\times G\to G:(x,y)\mapsto xy^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79da70aa0181309ff378473a661ea3ec45f6be83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.637ex; height:3.176ex;" alt="{\displaystyle G\times G\to G:(x,y)\mapsto xy^{-1}}"></span></dd></dl> <p>continu is.<sup id="cite_ref-sagle_1-0" class="reference"><a href="#cite_note-sagle-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Voorbeelden">Voorbeelden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topologische_groep&veaction=edit&section=1" title="Bewerk dit kopje: Voorbeelden" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topologische_groep&action=edit&section=1" title="De broncode bewerken van de sectie: Voorbeelden"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>De reële rechte voorzien van de <a href="/wiki/Optelling" class="mw-redirect" title="Optelling">optelling</a> is een <a href="/wiki/Abelse_groep" title="Abelse groep">abelse groep</a>. Hierbij is de inversie gewoon de tekenwisseling, of nog, de vermenigvuldiging met min een. Deze twee bewerkingen zijn continu voor de <a href="/w/index.php?title=Standaardtopologie&action=edit&redlink=1" class="new" title="Standaardtopologie (de pagina bestaat niet)">standaardtopologie</a> op de reële rechte. In het bijzonder hebben we een abelse topologische groep. Op een geheel analoge manier vormen de <a href="/wiki/Complexe_getallen" class="mw-redirect" title="Complexe getallen">complexe getallen</a> een abelse topologische groep voor de optelling.</li></ul> <ul><li>Beschouw de abelse groep <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} }_{0},.),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbb {R} }_{0},.),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ad9633b70a4071c27a39b76f87be6e87873628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.256ex; height:2.843ex;" alt="{\displaystyle ({\mathbb {R} }_{0},.),}"></span> de <a href="/wiki/Vermenigvuldigen" title="Vermenigvuldigen">vermenigvuldiging</a> van reële getallen verschillend van 0. De bewerkingen (de vermenigvuldiging en de multiplicatieve inversie) zijn continu voor de standaardtopologie. In het bijzonder hebben we te maken met een topologische groep. Op een geheel analoge manier kunnen we <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 {C} }_{0},.),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbb {C} }_{0},.),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2b9c0528d86f6560f024dcda297985aa98f799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.256ex; height:2.843ex;" alt="{\displaystyle ({\mathbb {C} }_{0},.),}"></span> het complexe vlak zonder de oorsprong voorzien van de vermenigvuldiging en de standaardtopologie, bekijken als een topologische groep.</li></ul> <ul><li>De vorige twee voorbeelden zijn allebei <a href="/wiki/Liegroep" class="mw-redirect" title="Liegroep">Liegroepen</a>. Elke Liegroep is vanzelf een topologische groep.</li></ul> <ul><li>Een afbeelding tussen <a href="/wiki/Discrete_ruimte" title="Discrete ruimte">discrete</a> topologische ruimten is altijd continu, dus een <a href="/wiki/Discrete_groep" title="Discrete groep">discrete groep</a> is een topologische groep.</li></ul> <ul><li>De triviale topologie <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 \{\emptyset ,G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo>,</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\emptyset ,G\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201a0db8e9b1b764427ecfb451fb8e879e6263ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.348ex; height:2.843ex;" alt="{\displaystyle \{\emptyset ,G\}}"></span> maakt van de groepsbewerking en de inversie continue afbeeldingen, maar voldoet niet aan de Hausdorff-eigenschap als <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> meer dan 1 element heeft.</li></ul> <ul><li>Een <a href="/wiki/Topologische_vectorruimte" title="Topologische vectorruimte">topologische vectorruimte</a> is een topologische groep voor de optelling van vectoren. De meeste auteurs nemen de Hausdorff-eigenschap op als onderdeel van de definitie van een topologische vectorruimte. Aangezien de studie van topologische vectorruimten meestal over <i>oneindigdimensionale</i> reële of complexe vectorruimten gaat, zijn dit geen Liegroepen in de traditionele zin.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Constructies">Constructies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topologische_groep&veaction=edit&section=2" title="Bewerk dit kopje: Constructies" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topologische_groep&action=edit&section=2" title="De broncode bewerken van de sectie: Constructies"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Merk op dat eenzelfde groep verschillende topologische structuren kan dragen en dus in principe aanleiding kan geven tot veel verschillende topologische groepen. Veronderstel bijvoorbeeld dat de groep <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"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> <mo>,</mo> </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/9bb996d4944a05f3316e34d284aa4a8ee8edafbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+),}"></span> de reële rechte van hierboven, is. Dan is dit zoals vermeld een topologische groep voor de standaardtopologie. Indien we echter de discrete topologie op <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"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </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/5b33b2c9358cbd7bad20aa0b18651d3bba582c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+)}"></span> leggen, verkrijgen we zoals net vermeld werd, ook een topologische groep. Deze objecten worden niet als equivalente topologische groepen beschouwd hoewel de onderliggende groepen duidelijk gelijk zijn. Voor de definitie van equivalentie, zie hieronder. Tenzij het duidelijk is welke topologie er op de groep <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 (G,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54a87abf331634c8962ef14c4c5ec41f94fd29c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}"></span> ligt, is het aangeraden ook de topologie <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> in de notatie te vermelden: <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 (G,T,*).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T,*).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1e175d40d8cbce41fc1c319a130c90dd1f8dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.149ex; height:2.843ex;" alt="{\displaystyle (G,T,*).}"></span> </p> <ul><li>Veronderstel dat <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 (G,T,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fbdb4fffb212c9c5e9927e34edee71ad1912d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:2.843ex;" alt="{\displaystyle (G,T,*)}"></span> een topologische groep is en dat <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> een deelgroep is van <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 G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}"></span> Indien <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> voorzien wordt van de <a href="/wiki/Deelruimtetopologie" title="Deelruimtetopologie">deelruimtetopologie</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 T_{H},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{H},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37a07a82fb8489ef92bc3eb2aa5be4f7d4ca2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.696ex; height:2.509ex;" alt="{\displaystyle T_{H},}"></span> dan is <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 (H,T_{H},*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (H,T_{H},*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7bc6de3db13931feb18ef745c6e8aee01e68fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.152ex; height:2.843ex;" alt="{\displaystyle (H,T_{H},*)}"></span> een topologische groep. Zo vormt de optelling van <a href="/wiki/Rationaal_getal" title="Rationaal getal">rationale getallen</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 {Q} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e00793ffac19dba314e2b6284b1dacf98f4c8ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} ,+)}"></span> een topologische deelgroep van <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"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </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/5b33b2c9358cbd7bad20aa0b18651d3bba582c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+)}"></span> die niet discreet is en ook geen Liegroep.</li></ul> <ul><li>Veronderstel dat <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 (G,T,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fbdb4fffb212c9c5e9927e34edee71ad1912d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:2.843ex;" alt="{\displaystyle (G,T,*)}"></span> een topologische groep is en dat <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> een <a href="/wiki/Normaaldeler" title="Normaaldeler">normale deelgroep</a> van <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is. Indien de <a href="/wiki/Factorgroep" title="Factorgroep">quotiëntgroep</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 G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}"></span> van <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> door <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> voorzien wordt van de <a href="/wiki/Quoti%C3%ABnttopologie" title="Quotiënttopologie">quotiënttopologie</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 T_{G/N},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{G/N},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7a667fefd665c35ac1c48fda1ba33c6386adc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.81ex; height:3.009ex;" alt="{\displaystyle T_{G/N},}"></span> dan is dit een topologische groep in de zin dat de groepsbewerking en de inversie continue afbeeldingen zijn. De Hausdorff-eigenschap voor de quotiëntruimte is gelijkwaardig met de eis dat de normale deelgroep ook <a href="/wiki/Gesloten_verzameling" title="Gesloten verzameling">gesloten</a> is in de topologie van <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 G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}"></span></li></ul> <ul><li>Het direct product van twee topologische groepen, gedefinieerd als de <a href="/wiki/Directe_som#Directe_som_van_twee_groepen" title="Directe som">directe som</a> van de groepen uitgerust met de <a href="/wiki/Producttopologie" title="Producttopologie">producttopologie</a>, is opnieuw een topologische groep.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Morfismen">Morfismen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topologische_groep&veaction=edit&section=3" title="Bewerk dit kopje: Morfismen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topologische_groep&action=edit&section=3" title="De broncode bewerken van de sectie: Morfismen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Een <b>morfisme</b> van een topologische groep <i>(G,T<sub>G</sub>,*)</i> naar een topologische groep <i>(H,T<sub>H</sub>,.)</i> is een morfisme van groepen van <i>G</i> naar <i>H</i> dat continu is voor de gegeven topologieen.</li></ul> <ul><li>Een <b>isomorfisme</b> van een topologische groep <i>(G,T<sub>G</sub>,*)</i> met een topologische groep <i>(H,T<sub>H</sub>,.)</i> is een morfisme van topologische groepen zodanig dat er een invers morfisme van topologische groepen bestaat. Concreet betekent dit: een <a href="/wiki/Isomorfisme" title="Isomorfisme">isomorfisme</a> van groepen dat tegelijkertijd een <a href="/wiki/Homeomorfisme" title="Homeomorfisme">homeomorfisme</a> van topologische ruimten is.</li></ul> <p>Beschouw nogmaals de twee voorbeelden van hierboven: <i>(R,+)</i> en <i>(R<sub>0</sub>,*)</i>, met de evidente topologie. De <a href="/wiki/Exponentieel" class="mw-redirect" title="Exponentieel">exponentiële afbeelding</a> van de eerste topologische groep naar de tweede, is een continu morfisme van groepen. De <a href="/wiki/Logaritme" title="Logaritme">logaritmische afbeelding</a> van de tweede naar de eerste, is ook een continu morfisme van groepen. Bovendien zijn deze exponentiële en logaritmische afbeeldingen elkaars inverse. In het bijzonder hebben we te maken met isomorfismen van topologische groepen. </p> <div class="mw-heading mw-heading2"><h2 id="Eigenschappen">Eigenschappen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topologische_groep&veaction=edit&section=4" title="Bewerk dit kopje: Eigenschappen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topologische_groep&action=edit&section=4" title="De broncode bewerken van de sectie: Eigenschappen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In een topologische groep zijn de topologische en de algebraïsche structuren compatibel. Het is dus te verwachten dat ze elkaar gaan beïnvloeden, en dit is ook zo. Beschouw bijvoorbeeld de volgende elementaire eigenschappen van topologische groepen: </p> <ul><li>De <a href="/wiki/Samenhang" title="Samenhang">samenhangende component</a> die het <a href="/wiki/Neutraal_element" title="Neutraal element">neutrale element</a> bevat, is een normale deelgroep die zelfs gesloten is.</li></ul> <ul><li>Topologische groepen zien er in elk punt hetzelfde uit: elke <a href="/wiki/Translatie_(meetkunde)" title="Translatie (meetkunde)">translatie</a> geeft een homeomorfisme van de groep. (Indien de groep niet abels is, hoeven de linker en de rechter translaties niet overeen te komen.) Ook de inversie geeft een homeomorfisme van de groep met zichzelf.</li></ul> <ul><li>Voor topologische groepen is de <a href="/wiki/Fundamentaalgroep" title="Fundamentaalgroep">fundamentaalgroep</a> altijd <a href="/wiki/Abelse_groep" title="Abelse groep">abels</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Noodzaak_van_de_Hausdorff-eigenschap">Noodzaak van de Hausdorff-eigenschap</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topologische_groep&veaction=edit&section=5" title="Bewerk dit kopje: Noodzaak van de Hausdorff-eigenschap" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topologische_groep&action=edit&section=5" title="De broncode bewerken van de sectie: Noodzaak van de Hausdorff-eigenschap"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De eis dat <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 (G,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0e4af15925a0fd2064b011847266b9ebb37637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.306ex; height:2.843ex;" alt="{\displaystyle (G,T)}"></span> een Hausdorff-ruimte (<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 T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>) is, kan zonder beperking worden verzwakt tot de <a href="/wiki/Scheidingsaxioma#T0:_Kolmogorov-ruimte" title="Scheidingsaxioma">Kolmogorov-eis <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 T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span></a>; er geldt zelfs:<sup id="cite_ref-sagle_1-1" class="reference"><a href="#cite_note-sagle-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>Zij <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 (G,T,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fbdb4fffb212c9c5e9927e34edee71ad1912d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:2.843ex;" alt="{\displaystyle (G,T,*)}"></span> een groep met continue bewerkingen die aan het Kolmogorov-axioma <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 T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span> voldoet. Dan is <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 (G,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0e4af15925a0fd2064b011847266b9ebb37637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.306ex; height:2.843ex;" alt="{\displaystyle (G,T)}"></span> ook <a href="/wiki/Scheidingsaxioma#T3½" title="Scheidingsaxioma">volledig regulier (<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 T_{3{1 \over 2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{3{1 \over 2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fca872fb1ca6187c018d3dff9f557eeed049f367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.915ex; height:4.009ex;" alt="{\displaystyle T_{3{1 \over 2}}}"></span>)</a> en dus zeker Hausdorff.</dd></dl> <dl><dd>Zij <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 (G,T,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fbdb4fffb212c9c5e9927e34edee71ad1912d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:2.843ex;" alt="{\displaystyle (G,T,*)}"></span> een <a href="/wiki/Lokaal_compacte_groep" title="Lokaal compacte groep">lokaal compacte groep</a> die aan het Kolmogorov-axioma <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 T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span> voldoet. Dan is <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 (G,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0e4af15925a0fd2064b011847266b9ebb37637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.306ex; height:2.843ex;" alt="{\displaystyle (G,T)}"></span> <a href="/wiki/Paracompacte_ruimte" title="Paracompacte ruimte">paracompact</a> en dus ook <a href="/wiki/Scheidingsaxioma#T4:_normale_ruimte@" title="Scheidingsaxioma">normaal (<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 T_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a8e4649fd334cf176ee7badff8eb949ecd3670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{4}}"></span>)</a>.</dd></dl> <p>Er bestaan topologische groepen die niet normaal zijn. </p> <div class="mw-heading mw-heading2"><h2 id="Zie_ook">Zie ook</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topologische_groep&veaction=edit&section=6" title="Bewerk dit kopje: Zie ook" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topologische_groep&action=edit&section=6" title="De broncode bewerken van de sectie: Zie ook"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Topologie" title="Topologie">Topologie</a> en <a href="/wiki/Topologische_ruimte" title="Topologische ruimte">topologische ruimte</a>.</li> <li><a href="/wiki/Groep_(wiskunde)" title="Groep (wiskunde)">Groepen</a> en <a href="/wiki/Groepentheorie" title="Groepentheorie">groepentheorie</a>.</li> <li><a href="/wiki/Compact" title="Compact">Compacte</a> groepen en <a href="/wiki/Lokaal_compacte_groep" title="Lokaal compacte groep">lokaal compacte groepen</a>, topologische groepen met enkele interessante topologische eigenschappen.</li> <li><a href="/wiki/Lie-groep" title="Lie-groep">Liegroepen</a>, groepen die niet alleen een topologie, maar zelfs een differentiaalstructuur toelaten.</li> <li><a href="/wiki/Topologische_vectorruimte" title="Topologische vectorruimte">Topologische vectorruimten</a> hebben een continue (en zelfs abelse) groepsbewerking, de optelling van vectoren.</li> <li><a href="/wiki/Morfisme" title="Morfisme">Morfismen</a>, <a href="/wiki/Isomorfisme" title="Isomorfisme">isomorfismen</a> en <a href="/wiki/Homeomorfisme" title="Homeomorfisme">homeomorfismen</a>.</li></ul> <div class="toccolours appendix" role="presentation" style="font-size:90%; margin:1em 0 -0.5em; clear:both;"> <div><span style="font-weight:bold">Bronnen, noten en/of referenties</span></div> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-sagle-1"><span class="mw-cite-backlink">↑ <a href="#cite_ref-sagle_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sagle_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Hoofdstuk 3 in Sagle, Arthur A. en Walde, Ralph E., "Introduction to Lie Groups and Lie Algebras," Pure and Applied Mathematics <b>51</b>, Academic Press 1973.</span> </li> </ol></div></div> </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="">Overgenomen van "<a dir="ltr" href="https://nl.wikipedia.org/w/index.php?title=Topologische_groep&oldid=54459985">https://nl.wikipedia.org/w/index.php?title=Topologische_groep&oldid=54459985</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categorie:Alles" title="Categorie:Alles">Categorieën</a>: <ul><li><a href="/wiki/Categorie:Topologie" title="Categorie:Topologie">Topologie</a></li><li><a href="/wiki/Categorie:Groepentheorie" title="Categorie:Groepentheorie">Groepentheorie</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Deze pagina is voor het laatst bewerkt op 29 aug 2019 om 16:49.</li> <li id="footer-info-copyright">De tekst is beschikbaar onder de licentie <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.nl">Creative Commons Naamsvermelding/Gelijk delen</a>, er kunnen aanvullende voorwaarden van toepassing zijn. 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