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<title>Isomorphismus</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Isomorphismus</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>In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist ein <b>Isomorphismus</b> (von <a href="Altgriechische_Sprache" title="Altgriechische Sprache">altgriechisch</a> <span lang="grc-Grek" class="Grek">ἴσος</span> (ísos) – „gleich“ und μορφή (morphḗ) – „Form“, „Gestalt“) eine Abbildung zwischen zwei <a href="Mathematische_Struktur" title="Mathematische Struktur">mathematischen Strukturen</a>, durch die Teile einer Struktur auf bedeutungsgleiche Teile einer anderen Struktur <a href="Bijektive_Funktion" title="Bijektive Funktion">umkehrbar eindeutig</a> <i>(bijektiv)</i> abgebildet werden.

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

<div class="mw-heading mw-heading3"><h3 id="Universelle_Algebra">Universelle Algebra</h3></div>

<p>In der <a href="Universelle_Algebra" title="Universelle Algebra">universellen Algebra</a> heißt eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</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 \varphi }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> zwischen zwei <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraischen Strukturen</a> (zum Beispiel <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a>, <a href="Ring_(Algebra)" title="Ring (Algebra)">Ringen</a>, <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körpern</a> oder <a href="Vektorraum" title="Vektorraum">Vektorräumen</a>) ein <i>Isomorphismus</i>, wenn:

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> <a href="Bijektive_Funktion" title="Bijektive Funktion">bijektiv</a> ist,</li>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> ein <a href="Homomorphismus" title="Homomorphismus">Homomorphismus</a> ist.</li></ul>

<p>Gibt es einen Isomorphismus zwischen zwei algebraischen Strukturen, dann heißen die beiden Strukturen zueinander <i>isomorph</i>. Isomorphe Strukturen sind in gewisser Weise „das gleiche“, nämlich dann, wenn man von der Darstellung der Elemente der zugrundeliegenden Mengen und den Namen der Relationen und Verknüpfungen absieht.

</p><p>Die Aussage „<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> 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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> sind isomorph“ wird üblicherweise 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 \simeq }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65b9738551241417d16d9843525ed52410af4dc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \simeq }" loading="lazy"></span> oder 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 X\cong Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2a962b1ee3f7c2dd1b3ae636037f4c947107190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X\cong Y}" loading="lazy"></span> notiert.

</p><p>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 \varphi }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> ein bijektiver Homomorphismus zwischen zwei algebraischen Strukturen, dann ist immer auch <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 \varphi ^{-1}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5393676141f43c07be34b09aeb65f22064e92cf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.853ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}}" loading="lazy"></span> ein bijektiver Homomorphismus.

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<div class="mw-heading mw-heading3"><h3 id="Relationale_Strukturen">Relationale Strukturen</h3></div>

<p>Es seien <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 {\boldsymbol {A}}=(A,(R_{i}))}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}=(A,(R_{i}))}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d192e1a883a5c6d8b08fb03b2cf875434efd065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.077ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {A}}=(A,(R_{i}))}" 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 {\boldsymbol {B}}=(B,(S_{i}))}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {B}}=(B,(S_{i}))}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4fb65c5170b75c7dcbe5c0e70dd92494f068c80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.752ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {B}}=(B,(S_{i}))}" loading="lazy"></span> zwei <a href="Relationale_Struktur" class="mw-redirect" title="Relationale Struktur">relationale Strukturen</a> vom gleichen Typ <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_{i}),}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a381973ee157fc01242077f7dd3772ffebc26f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.651ex; height:2.843ex;" alt="{\displaystyle (n_{i}),}" loading="lazy"></span> sodass <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_{i}\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n_{i}\in \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17cc13c5962f46501fd1d792ac09ab2fd6f2527d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.713ex; height:2.509ex;" alt="{\displaystyle n_{i}\in \mathbb {N} }" loading="lazy"></span> für jedes <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 i}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> die <a href="Stelligkeit" title="Stelligkeit">Stelligkeit der Relationen</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 R_{i}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db421291be9d0103404ced7495b363437b67b6b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle R_{i}}" 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 S_{i}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de6e810a93f67802ecb603ee0e3324005c6e583e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}" loading="lazy"></span> bezeichnet. Eine <a href="Bijektion" class="mw-redirect" title="Bijektion">Bijektion</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 \varphi \colon A\to B}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \colon A\to B}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61676cfcb7acc6bc6d6f13a228ffaaf0a879e8c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.675ex; height:2.676ex;" alt="{\displaystyle \varphi \colon A\to B}" loading="lazy"></span> heißt <i>Isomorphismus</i>, wenn sie für jedes <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 i}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> und für alle <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_{1},\ldots ,a_{n_{i}}\in A}">

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<annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n_{i}}\in A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/378f416b8088de6cc19183c45180cf0c09e8f2c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.119ex; height:2.843ex;" alt="{\displaystyle a_{1},\ldots ,a_{n_{i}}\in A}" loading="lazy"></span> die folgende <i>Verträglichkeitseigenschaft</i> besitzt:

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<annotation encoding="application/x-tex">{\displaystyle (a_{1},\ldots ,a_{n_{i}})\in R_{i}\Leftrightarrow (\varphi (a_{1}),\ldots ,\varphi (a_{n_{i}}))\in S_{i}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9ca6821babc22ebb3a304723aed59910c80fb03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.079ex; height:3.009ex;" alt="{\displaystyle (a_{1},\ldots ,a_{n_{i}})\in R_{i}\Leftrightarrow (\varphi (a_{1}),\ldots ,\varphi (a_{n_{i}}))\in S_{i}.}" loading="lazy"></span></dd></dl>

<p>Im Gegensatz zu algebraischen Strukturen ist nicht jeder bijektive Homomorphismus zwischen relationalen Strukturen ein Isomorphismus. Ein Beispiel für Isomorphismen zwischen relationalen Strukturen sind <a href="Isomorphie_von_Graphen" title="Isomorphie von Graphen">Isomorphismen zwischen Graphen</a>.

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

<p>In der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a> definiert man einen Isomorphismus allgemein als einen <a href="Morphismus" title="Morphismus">Morphismus</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 f\colon X\to Y,}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49d011f18f93c8dd315e1239145fa13330ef1f25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.327ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y,}" loading="lazy"></span> der ein beidseitiges Inverses <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}\colon \,Y\to X}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}\colon \,Y\to X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9ebbdca87309d737ee10b090af0bf033ca1a384.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.442ex; height:3.009ex;" alt="{\displaystyle f^{-1}\colon \,Y\to X}" loading="lazy"></span> besitzt:

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<annotation encoding="application/x-tex">{\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7aa98f6bfb3b3dd459b72b68ad61efe1a78f1c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.651ex; height:3.009ex;" alt="{\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}}" 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 f^{-1}\circ f=\operatorname {id} _{X}.}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}\circ f=\operatorname {id} _{X}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d53f089991fab2e298f0478aef3c1e5cba110c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.444ex; height:3.009ex;" alt="{\displaystyle f^{-1}\circ f=\operatorname {id} _{X}.}" loading="lazy"></span></dd></dl>

<p>Die oben definierten Isomorphismen zwischen algebraischen Strukturen sowie zwischen relationalen Strukturen sind Spezialfälle dieser Definition. Weitere Spezialfälle dieses Isomorphiebegriffes sind beispielsweise <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismen</a> als Isomorphismen in der Kategorie der <a href="Topologischer_Raum" title="Topologischer Raum">topologischen Räume</a> und <a href="Stetige_Funktion" title="Stetige Funktion">stetige Abbildungen</a> oder <a href="Homotopie#Homotopieäquivalenz" title="Homotopie">Homotopieäquivalenzen</a> als Isomorphismen in der Kategorie der topologischen Räume mit den <a href="Homotopie" title="Homotopie">Homotopieklassen</a> von Abbildungen als Morphismen.

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

<p>In der Kategorientheorie ist von entscheidender Bedeutung, dass die Eigenschaft Isomorphismus unter jedem <a href="Funktor_(Mathematik)#Elementare_Eigenschaften" title="Funktor (Mathematik)">Funktor</a> erhalten bleibt, d. h. 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\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"></span> ein Isomorphismus in einer Kategorie <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 C}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" 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 F\colon C\to D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\colon C\to D}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8299d81a3d96658bf9240b799a248872f36a3772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.079ex; height:2.176ex;" alt="{\displaystyle F\colon C\to D}" loading="lazy"></span> ein Funktor, dann ist

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(f)\colon F(X)\to F(Y)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6706e8089dac8a824056309cb3cdb49919f543cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.33ex; height:2.843ex;" alt="{\displaystyle F(f)\colon F(X)\to F(Y)}" loading="lazy"></span></dd></dl>

<p>ebenfalls ein Isomorphismus in der Kategorie <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 D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>. In der <a href="Algebraische_Topologie" title="Algebraische Topologie">algebraischen Topologie</a> wird diese Eigenschaft häufig festgestellt, um Räume in Relation bringen zu können: Sind beispielsweise zwei Räume <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">homöomorph</a>, so sind ihre <a href="Fundamentalgruppe" title="Fundamentalgruppe">Fundamentalgruppen</a> isomorph.

</p>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed85ceae5a6c5fbb0c42f9a0ad7525301582696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.47ex; height:2.843ex;" alt="{\displaystyle (X,\cdot )}" 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 \left(Y,+\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(Y,+\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb02c070e181e23546a305d224b89df28c668866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.425ex; height:2.843ex;" alt="{\displaystyle \left(Y,+\right)}" loading="lazy"></span> Mengen mit einer <a href="Zweistellige_Verkn%C3%BCpfung" title="Zweistellige Verknüpfung">binären Verknüpfung</a>, dann ist eine <a href="Bijektion" class="mw-redirect" title="Bijektion">Bijektion</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 f\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"></span> mit

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(u)+f(v)=f(u\cdot v)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fabdf7473f4e7b915a446b177aa9e44c74f00e43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.796ex; height:2.843ex;" alt="{\displaystyle f(u)+f(v)=f(u\cdot v)}" loading="lazy"></span> für alle <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 u,v\in X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44d42de074fa8552007d76aa1842b9cbbf25215e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.312ex; height:2.509ex;" alt="{\displaystyle u,v\in X}" loading="lazy"></span></dd></dl>

<p>ein Isomorphismus 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> 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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>. So ist etwa der Logarithmus ein Isomorphismus 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 (\mathbb {R} ^{+},/)}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{+},/)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/782fd7cd8c6c5c1426e5df9fa2c094cf151919e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.195ex; height:3.009ex;" alt="{\displaystyle (\mathbb {R} ^{+},/)}" 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 (\mathbb {R} ,-)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbbcd71acbb8bd8703d77ae64e2ef52e5d614a35.svg" 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} ,-)}" loading="lazy"></span>, da <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 \log(x)-\log(y)=\log \left({\tfrac {x}{y}}\right)}">

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log(x)-\log(y)=\log \left({\tfrac {x}{y}}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/341bdbf56da41433e1e4cb72a5d9eb8fce81f270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.51ex; height:4.843ex;" alt="{\displaystyle \log(x)-\log(y)=\log \left({\tfrac {x}{y}}\right)}" loading="lazy"></span>.

</p><p>Eine binäre Verknüpfung ist eine dreistellige Relation. Aber auch zu zweistelligen <a href="Relation_(Mathematik)" title="Relation (Mathematik)">Relationen</a> lassen sich Homo- und Isomorphismen definieren (s. u. <a href="#Ordnungsisomorphismus">#Ordnungsisomorphismus</a>).

</p><p>Bei manchen Isomorphismen impliziert die Homomorphie der Funktion auch die der Umkehrfunktion; bei den anderen muss man sie extra nachweisen.

</p>

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

<p>Sind die Strukturen <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a>, dann heißt ein solcher Isomorphismus <a href="Gruppenisomorphismus" title="Gruppenisomorphismus">Gruppenisomorphismus</a>. Meist meint man mit Isomorphismen solche zwischen algebraischen Strukturen wie Gruppen, <a href="Ring_(Algebra)" title="Ring (Algebra)">Ringen</a>, <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körpern</a> oder <a href="Vektorraum" title="Vektorraum">Vektorräumen</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Isometrischer_Isomorphismus">Isometrischer Isomorphismus</h3></div>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(X,d\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c64a98f2d81cceafa7d7e556e5efc48617513a7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle \left(X,d\right)}" 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 \left(Y,D\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(Y,D\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db9c15e3e7627244e6c72a22d59efbd3fac598ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.541ex; height:2.843ex;" alt="{\displaystyle \left(Y,D\right)}" loading="lazy"></span> <a href="Metrischer_Raum" title="Metrischer Raum">metrische Räume</a> und 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> eine Bijektion 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> 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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> mit der Eigenschaft

</p>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\left(f(u),f(v)\right)=d(u,v)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2669ea342233a62a98c1dd2d9acec5cecbb0a3dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.403ex; height:2.843ex;" alt="{\displaystyle D\left(f(u),f(v)\right)=d(u,v)}" loading="lazy"></span> für alle <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 u,v\in X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44d42de074fa8552007d76aa1842b9cbbf25215e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.312ex; height:2.509ex;" alt="{\displaystyle u,v\in X}" loading="lazy"></span>,</dd></dl>

<p>dann nennt man <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> einen <i>isometrischen Isomorphismus</i>.

</p><p>In den bisherigen Beispielen sind Isomorphismen genau die homomorphen Bijektionen – die Umkehrabbildung ist automatisch homomorph. In den folgenden Beispielen muss zusätzlich gefordert werden, dass auch die Umkehrabbildung homomorph ist.

</p><p>In der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a> nennt man eine Abbildung <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 T\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83d566b82c1cdd7151aa7786c447d6c3cc4f5997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.038ex; height:2.176ex;" alt="{\displaystyle T\colon X\to Y}" loading="lazy"></span> zwischen <a href="Normierter_Raum" title="Normierter Raum">normierten Räumen</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,\|\cdot \|_{X}),(Y,\|\cdot \|_{Y})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\|\cdot \|_{X}),(Y,\|\cdot \|_{Y})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10a53845cfe5b73509daf40cba889c33dddc2e96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.6ex; height:2.843ex;" alt="{\displaystyle (X,\|\cdot \|_{X}),(Y,\|\cdot \|_{Y})}" loading="lazy"></span> einen Isomorphismus, wenn sie folgende Eigenschaften hat:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" 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}" loading="lazy"></span> ist <a href="Lineare_Abbildung" title="Lineare Abbildung">linear</a></li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" 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}" loading="lazy"></span> ist <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a></li>

<li>Die <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</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 T^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a1a5bed0ce2d8adc7fdad412c34ef905fb5026f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.053ex; height:2.676ex;" alt="{\displaystyle T^{-1}}" loading="lazy"></span> ist auch <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a></li></ul>

<p>Falls zusätzlich für alle <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}">

</mstyle>

</mrow>

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

</semantics>

</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> gilt <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 \|T(x)\|_{Y}=\|x\|_{X}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|T(x)\|_{Y}=\|x\|_{X}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f38174d0bfb12d5773a263ec0e5665e4c58ae0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.972ex; height:2.843ex;" alt="{\displaystyle \|T(x)\|_{Y}=\|x\|_{X}}" loading="lazy"></span>, so nennt man <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 T}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" 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}" loading="lazy"></span> einen <a href="Isometrische_Isomorphie" title="Isometrische Isomorphie">isometrischen Isomorphismus</a>.

</p>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc8b6ab25f3e3e993bb5b670611d6595c5b48aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.264ex; height:2.843ex;" alt="{\displaystyle (X,\leq _{X})}" 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 (Y,\leq _{Y})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Y,\leq _{Y})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/241a2d38f8b72ef62a93824b2ac07a5b61bc27f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.911ex; height:2.843ex;" alt="{\displaystyle (Y,\leq _{Y})}" loading="lazy"></span> <a href="Ordnungsrelation" title="Ordnungsrelation">geordnete</a> Mengen, dann ist ein (Ordnungs-)Isomorphismus 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> 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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> eine ordnungserhaltende Bijektion, deren Umkehrfunktion ebenfalls ordnungserhaltend ist. Ordnungserhaltende Bijektionen zwischen totalgeordneten Mengen sind automatisch Isomorphismen; für Halbordnungen gilt dies nicht: <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\mapsto n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb68960a2b3bac8a6f2d5223fc1f45c2edae1175.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.404ex; height:1.843ex;" alt="{\displaystyle n\mapsto n}" loading="lazy"></span> ist offenkundig eine ordnungserhaltende Bijektion 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 \mathbb {N} ^{+}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a339dbd4a64016f4d222b5cd5d840d77041924a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} ^{+}}" loading="lazy"></span> mit der Teilerrelation 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 \mathbb {N} ^{+}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a339dbd4a64016f4d222b5cd5d840d77041924a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} ^{+}}" loading="lazy"></span> mit der gewöhnlichen Ordnung, aber nicht in der Gegenrichtung.

Ordnungsisomorphismen spielen in der Theorie der <a href="Ordinalzahlen" class="mw-redirect" title="Ordinalzahlen">Ordinalzahlen</a> eine wichtige Rolle.

Man sagt auch, <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> 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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> seien <i>ordnungsisomorph</i> oder vom selben <a href="Ordnungstyp" class="mw-redirect" title="Ordnungstyp">Ordnungstyp</a>.

Der Ordnungstyp der natürlichen Zahlen <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 {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf9a96b565ea202d0f4322e9195613fb26a9bed.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 {N} }" loading="lazy"></span> wird mit <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 \omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> und der der rationalen Zahlen <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 {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"></span> mit <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 \eta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> bezeichnet. Der Ordnungstyp der rationalen Zahlen im offenen Intervall <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 \{q\in \mathbb {Q} \mid 0<q<1\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{q\in \mathbb {Q} \mid 0<q<1\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9471684fe60511211593517901ac8034cc4e368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.572ex; height:2.843ex;" alt="{\displaystyle \{q\in \mathbb {Q} \mid 0<q<1\}}" loading="lazy"></span> ist ebenfalls <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 \eta .}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc94fc42a3ecbad87643808e17ec9634147cf812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.816ex; height:2.176ex;" alt="{\displaystyle \eta .}" loading="lazy"></span> Beide sind <a href="Dichte_Teilmenge" title="Dichte Teilmenge">dicht</a> in ihrer jeweiligen <a href="Vollst%C3%A4ndiger_Raum#Vervollständigung" title="Vollständiger Raum">Vervollständigung</a>. Die Ordnungstypen der reellen Zahlen <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> und des Intervalls <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 (0,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy"></span> sind ebenfalls gleich, aber verschieden 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 \eta ,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58df040ac1b922ab90bec04b75b9ac3156d544ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.816ex; height:2.176ex;" alt="{\displaystyle \eta ,}" loading="lazy"></span> da es keine Bijektion zwischen <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> 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 \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"></span> gibt.

</p>

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

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

<li><a href="Automorphismus" title="Automorphismus">Automorphismus</a></li>

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

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

<ul><li><a href="Klaus_J%C3%A4nich" title="Klaus Jänich">Klaus Jänich</a>: <i>Topologie.</i> 8. Auflage, 1. korrigierter Nachdruck. Springer, Berlin u. a. 2006, ISBN 3-540-21393-7.</li></ul>

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

<div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"></span></span></span><b><a href="https://de.wiktionary.org/wiki/Isomorphismus" class="extiw external" title="wikt:Isomorphismus">Wiktionary: Isomorphismus</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div></div><div><div class="zim-footer">

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