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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Subbasis</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Eine <b>Subbasis</b> ist in der mathematischen Grundlagendisziplin der <a href="Mengentheoretische_Topologie" class="mw-redirect" title="Mengentheoretische Topologie">mengentheoretischen Topologie</a> ein spezielles <a href="Mengensystem" title="Mengensystem">Mengensystem</a> von <a href="Offene_Menge" title="Offene Menge">offenen Mengen</a>. Eine Subbasis bestimmt eine <a href="Topologischer_Raum" title="Topologischer Raum">Topologie</a> eindeutig und vereinfacht damit oftmals Beweise, da es ausreichend ist, sich auf die Mengen der Subbasis zu beschränken. Ebenso werden manche Eigenschaften von Topologien auch als Eigenschaften ihrer Subbasen definiert.

</p><p>Umgekehrt lässt sich jedes Mengensystem als Subbasis auffassen und ermöglicht es so, gezielt Topologien mit bestimmten Eigenschaften zu konstruieren.

</p><p>In der aus dem Russischen ins Englische übersetzten Literatur findet sich auch die Bezeichnung „Pre-Base“ (deutsch: <i>Prä-Basis</i>) anstelle der typischen englischen Bezeichnungen <i>subbase</i> oder <i>subbasis</i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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

<p>Gegeben sei ein <a href="Topologischer_Raum" title="Topologischer Raum">topologischer Raum</a> <span class="mwe-math-element mwe-math-element-inline"><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 {O}})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/922442a5575888b509d846450a3fcf1ae09c2f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.673ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}})}" loading="lazy"></span> sowie ein <a href="Mengensystem" title="Mengensystem">Mengensystem</a> <span class="mwe-math-element mwe-math-element-inline"><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 {S}}\subset {\mathcal {O}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f01c8a81ac5159a7b1a98f7829b25e8f09c3bfa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.441ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}\subset {\mathcal {O}}}" loading="lazy"></span>. Dann heißt <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> eine Subbasis der Topologie <span class="mwe-math-element mwe-math-element-inline"><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 {O}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span>, wenn eine der folgenden äquivalenten Bedingungen erfüllt ist:

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<ul><li>Jede offene Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\in {\mathcal {O}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c1463be4e96b8ec525d382b721b3db573b5f1c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.464ex; height:2.176ex;" alt="{\displaystyle O\in {\mathcal {O}}}" loading="lazy"></span> ist die Vereinigung von beliebig vielen Mengen, die selbst <a href="Menge_(Mathematik)#Durchschnitt_(Schnittmenge,_Schnitt)" title="Menge (Mathematik)">Durchschnitte</a> von endlich vielen Mengen aus <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> sind.</li>

<li>Die Menge aller Durchschnitte von endlich vielen Mengen aus <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>, also</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}:=\left\lbrace M\subset X\,|\,M=\bigcap _{j\in J}S_{j},\;S_{j}\in {\mathcal {S}},\;|J|<\infty \right\rbrace }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1da5415084b7207300932839b3d6b59f75fdb6d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.266ex; height:7.676ex;" alt="{\displaystyle {\mathcal {B}}:=\left\lbrace M\subset X\,|\,M=\bigcap _{j\in J}S_{j},\;S_{j}\in {\mathcal {S}},\;|J|<\infty \right\rbrace }" loading="lazy"></span></dd>

<dd>bildet eine <a href="Basis_(Topologie)" title="Basis (Topologie)">Basis</a> der Topologie <span class="mwe-math-element mwe-math-element-inline"><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 {O}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span>.</dd></dl>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> erzeugt <span class="mwe-math-element mwe-math-element-inline"><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 {O}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span> in dem Sinne, dass</li></ul>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span> die (bezüglich Teilmengenbeziehung) kleinste Topologie ist, die <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> enthält, und</li>

<li>jede weitere Topologie, die <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> enthält, immer <a href="Gr%C3%B6bere_und_feinere_Topologien" title="Gröbere und feinere Topologien">feiner</a> ist als <span class="mwe-math-element mwe-math-element-inline"><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 {O}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span>.</li></ul></dd></dl>

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

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Unendliche_Menge" title="Unendliche Menge">unendliche Menge</a>, so bildet die Menge aller endlichen Teilmengen einer vorgegebenen, endlichen Mächtigkeit <span class="mwe-math-element mwe-math-element-inline"><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\neq 0}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5920e98ff3dd1cb41e01f76243300450c958d5e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.676ex;" alt="{\displaystyle n\neq 0}" loading="lazy"></span>, also

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}:=\{M\subset X\,|\,|M|=n\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fa50df6ed17b7f1183b266b1487cefacaa6bbab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.733ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}:=\{M\subset X\,|\,|M|=n\}}" loading="lazy"></span></dd></dl>

<p>eine Subbasis der <a href="Diskrete_Topologie" title="Diskrete Topologie">diskreten Topologie</a>, die durch <span class="mwe-math-element mwe-math-element-inline"><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 {O}}_{D}:={\mathcal {P}}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{D}:={\mathcal {P}}(X)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8269589fe976aca517f001a373ff65d3f9911434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.681ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{D}:={\mathcal {P}}(X)}" loading="lazy"></span> gegeben ist. Denn es gilt nach Auswahl geeigneter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1},S_{2}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05289e5891f97d54bdc946771362aedd6bb76917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.992ex; height:2.509ex;" alt="{\displaystyle S_{1},S_{2}}" loading="lazy"></span> aus <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span>, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}\cap S_{2}=\{x\}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa56f3d2a0438fe02eea08750f231c579ff41b69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.294ex; height:2.843ex;" alt="{\displaystyle S_{1}\cap S_{2}=\{x\}}" loading="lazy"></span> für ein vorgegebenes <span class="mwe-math-element mwe-math-element-inline"><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 X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>. Somit lassen sich aus <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> alle einelementigen Teilmengen von <span class="mwe-math-element mwe-math-element-inline"><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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> erzeugen. Diese bilden dann eine Basis der diskreten Topologie.

</p><p>Eine Subbasis der natürlichen Topologie auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> ist gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}:={\mathcal {S}}^{+}\cup {\mathcal {S}}^{-}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2ad67da34710214c4edad7d681439b48fb972e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.877ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}:={\mathcal {S}}^{+}\cup {\mathcal {S}}^{-}}" loading="lazy"></span>,</dd></dl>

<p>wobei

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{-}:=\{(-\infty ,b)\,|\,b\in \mathbb {R} \}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ffa067fb91a8d9d455af916dcde36a53e12474e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.008ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}^{-}:=\{(-\infty ,b)\,|\,b\in \mathbb {R} \}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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 {S}}^{+}:=\{(a,+\infty )\,|\,a\in \mathbb {R} \}}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{+}:=\{(a,+\infty )\,|\,a\in \mathbb {R} \}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fae2cd4b45af32bc42a21c264df546850581de8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.473ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}^{+}:=\{(a,+\infty )\,|\,a\in \mathbb {R} \}}" loading="lazy"></span></dd></dl>

<p>ist. Denn die Menge der offenen Intervalle bildet eine Basis der natürlichen Topologie, und jedes offene Intervall lässt sich aus der Subbasis durch

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)=(-\infty ,b)\cap (a,+\infty )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7f983924919b2fffca8d632777442f3d116768e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.929ex; height:2.843ex;" alt="{\displaystyle (a,b)=(-\infty ,b)\cap (a,+\infty )}" loading="lazy"></span></dd></dl>

<p>erzeugen.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Nicht-Eindeutigkeit">Nicht-Eindeutigkeit</h3></div>

<p>Subbasen bestimmen zwar die Topologie eindeutig, im Allgemeinen besitzt eine Topologie aber mehr als eine Subbasis. So bilden sowohl

</p>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}:=\{\{1\},\{2\},\{3\}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/282067fab099d5d960b85b77053b084dda9db93a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.063ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}_{1}:=\{\{1\},\{2\},\{3\}\}}" loading="lazy"></span> als auch</dd>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{2}:=\{\{1,2\},\{2,3\},\{3,1\}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/859d71e181bb99d1a52ce49603cb848f6379219d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.652ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}_{2}:=\{\{1,2\},\{2,3\},\{3,1\}\}}" loading="lazy"></span></dd></dl>

<p>eine Subbasis von <span class="mwe-math-element mwe-math-element-inline"><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 {O}}={\mathcal {P}}(\{1,2,3\})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}={\mathcal {P}}(\{1,2,3\})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a878beaf02ebc46ddf1cfd6f20bb1d90f05fe7d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.341ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}={\mathcal {P}}(\{1,2,3\})}" loading="lazy"></span>. Ebenso besitzt die natürliche Topologie auf <span class="mwe-math-element mwe-math-element-inline"><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} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> nicht bloß die oben als Beispiel angegebene Subbasis. Es genügt beispielsweise auch, Intervalle der Form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,a)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c95e49fd3f33ee3f4bb536214d7860f0eab27577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle (-\infty ,a)}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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,+\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b,+\infty )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/490ba7cec8e0b6c3a169bc2608036a8f5ea759f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.973ex; height:2.843ex;" alt="{\displaystyle (b,+\infty )}" loading="lazy"></span> für rationale Intervallgrenzen, also für <span class="mwe-math-element mwe-math-element-inline"><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\in \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a00cbe702ec1d24ff0707e2a618290801a95856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.91ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {Q} }" loading="lazy"></span> zu betrachten.

</p>

<div class="mw-heading mw-heading3"><h3 id="Erzeugung_von_Topologien_durch_Subbasen">Erzeugung von Topologien durch Subbasen</h3></div>

<p>So wie eine Topologie ihre Subbasen bestimmt, kann man ebenso durch eine Subbasis eine Topologie bestimmen. Dazu wählt man ein beliebiges Mengensystem <span class="mwe-math-element mwe-math-element-inline"><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 {M}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" loading="lazy"></span> und erklärt dies zur Subbasis einer vorerst nicht näher präzisierten Topologie. Zu beachten ist hier, dass dies im Gegensatz zum analogen Verfahren mit Basen ohne jegliche Voraussetzung an das Mengensystem möglich ist.

</p><p>Formell wird dieses Verfahren, das sich in der dritten der oben gegebenen Definitionen widerspiegelt, durch den <a href="H%C3%BCllenoperator" title="Hüllenoperator">Hüllenoperator</a>

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mtext> ist Topologie auf </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau ({\mathcal {M}}):=\bigcap \{{\mathcal {E}}\subseteq {\mathcal {P}}(X)\,|\,{\mathcal {M}}\subseteq {\mathcal {E}},\,{\mathcal {E}}{\text{ ist Topologie auf }}X\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c028c3e00456c8ead242a6c56688cfcaa6559fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:55.593ex; height:3.843ex;" alt="{\displaystyle \tau ({\mathcal {M}}):=\bigcap \{{\mathcal {E}}\subseteq {\mathcal {P}}(X)\,|\,{\mathcal {M}}\subseteq {\mathcal {E}},\,{\mathcal {E}}{\text{ ist Topologie auf }}X\}}" loading="lazy"></span>.</dd></dl>

<p>Dieser Hüllenoperator liefert wieder eine Topologie, da der Schnitt von Topologien wieder eine Topologie ist. Des Weiteren ist diese Topologie die gröbste Topologie, die das vorgegebene Mengensystem <span class="mwe-math-element mwe-math-element-inline"><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 {M}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" loading="lazy"></span> enthält.

</p>

<div class="mw-heading mw-heading2"><h2 id="Wichtige_Aussagen_mittels_Subbasen">Wichtige Aussagen mittels Subbasen</h2></div>

<ul><li>Die <a href="Initialtopologie" title="Initialtopologie">Initialtopologie</a> einer Familie von Abbildungen <span class="mwe-math-element mwe-math-element-inline"><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_{i})_{i\in I}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i\in I}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f3dd9718b5e3460904d390a0d092237aec266ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.473ex; height:2.843ex;" alt="{\displaystyle (f_{i})_{i\in I}}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> in die topologischen Räume <span class="mwe-math-element mwe-math-element-inline"><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_{i},{\mathcal {O}}_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Y_{i},{\mathcal {O}}_{i})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b67090992ec8f4ad3cd9269429921275a7332a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.643ex; height:2.843ex;" alt="{\displaystyle (Y_{i},{\mathcal {O}}_{i})}" loading="lazy"></span> ist genau die Topologie auf <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, deren Subbasis aus den Urbildern offener Mengen, also aus <span class="mwe-math-element mwe-math-element-inline"><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_{i}^{-1}(O_{i})}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d02bbd077b51f6582d6383ec43abbb3afef7b70b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.036ex; height:3.343ex;" alt="{\displaystyle f_{i}^{-1}(O_{i})}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{i}\in {\mathcal {O}}_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58d54cf7fd66f81edd8ede3c88e9214bb4894d58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.063ex; height:2.509ex;" alt="{\displaystyle O_{i}\in {\mathcal {O}}_{i}}" loading="lazy"></span>, besteht. Da sowohl die <a href="Teilraumtopologie" title="Teilraumtopologie">Teilraumtopologie</a> als auch die <a href="Produkttopologie" title="Produkttopologie">Produkttopologie</a> Spezialfälle der Initialtopologie sind, lassen sich diese Topologien ebenso über ihre Subbasen definieren.</li>

<li><a href="Satz_von_Alexander_(Mengentheoretische_Topologie)" class="mw-redirect" title="Satz von Alexander (Mengentheoretische Topologie)">Satz von Alexander</a>: Es genügt, <a href="Kompakter_Raum" title="Kompakter Raum">Kompaktheit</a> für Mengen aus einer Subbasis zu überprüfen.</li>

<li>Ebenfalls genügt es, <a href="Stetige_Funktion" title="Stetige Funktion">Stetigkeit</a> auf einer Subbasis zu überprüfen. Ist also <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> eine Abbildung von <span class="mwe-math-element mwe-math-element-inline"><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_{1},{\mathcal {O}}_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X_{1},{\mathcal {O}}_{1})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85bd967e43fdb861e1b7bbf1fab5476ed4e6d14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle (X_{1},{\mathcal {O}}_{1})}" loading="lazy"></span> nach <span class="mwe-math-element mwe-math-element-inline"><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_{2},{\mathcal {O}}_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X_{2},{\mathcal {O}}_{2})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/633fedb639ac1d7323bb4fe47f794d698c241f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle (X_{2},{\mathcal {O}}_{2})}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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 {S}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> eine beliebige Subbasis von <span class="mwe-math-element mwe-math-element-inline"><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 {O}}_{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70fc7bef63e5ae0ca1e2efa7d5161bc9d0215b4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.904ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{2}}" loading="lazy"></span>, so ist <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> genau dann stetig, wenn <span class="mwe-math-element mwe-math-element-inline"><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}({\mathcal {S}})\subset {\mathcal {O}}_{1}}">

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}({\mathcal {S}})\subset {\mathcal {O}}_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9675cce71b5ba40f81bc9dcd2fda8a5d826207e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.957ex; height:3.176ex;" alt="{\displaystyle f^{-1}({\mathcal {S}})\subset {\mathcal {O}}_{1}}" loading="lazy"></span> ist.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

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

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

<ul><li>Margherita Barile: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Subbasis.html"><i>Subbasis</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul>

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

<ul><li>Steven Roman: <cite style="font-style:italic">Lattices and Ordered Sets</cite>. Springer, 2008, ISBN 978-0-387-78900-2, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-0-387-78901-9">10.1007/978-0-387-78901-9</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Subbasis&rft.au=Steven+Roman&rft.btitle=Lattices+and+Ordered+Sets&rft.date=2008&rft.doi=10.1007%2F978-0-387-78901-9&rft.genre=book&rft.isbn=9780387789002&rft.pub=Springer" style="display:none"> </span></li>

<li><a href="Boto_von_Querenburg" title="Boto von Querenburg">Boto von Querenburg</a>: <cite style="font-style:italic">Mengentheoretische Topologie</cite>. 3. Auflage. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-56860-2">10.1007/978-3-642-56860-2</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Subbasis&rft.au=Boto+von+Querenburg&rft.btitle=Mengentheoretische+Topologie&rft.date=2001&rft.doi=10.1007%2F978-3-642-56860-2&rft.edition=3.&rft.genre=book&rft.isbn=9783540677901&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none"> </span></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">M.I. Voitsekhovskii: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Pre-Base</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Pre-base">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Subbasis&rft.atitle=Pre-Base&rft.au=M.I.+Voitsekhovskii&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span> </span>

</li>

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