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width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>) bezeichnet. Ein <i>(beschränktes)</i> Intervall besteht aus allen Elementen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, die man mit zwei begrenzenden Elementen der Trägermenge (<b>Intervallgrenzen</b>), der <i>unteren Grenze</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und der <i>oberen Grenze <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> des Intervalls,</i> der Größe nach vergleichen kann und die im Sinne dieses Vergleichs <i>zwischen</i> den Grenzen liegen. Dabei können die Grenzen des Intervalls dem Intervall angehören <i>(abgeschlossenes Intervall, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq x\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq x\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8725e3444a04e54561699c3255f7489e22b63a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.754ex; height:2.343ex;" alt="{\displaystyle a\leq x\leq b}"></span>)</i>, nicht angehören <i>(offenes Intervall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<x<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<x<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09d6f5cf7cd5aab0408e75dae5f10109ae9fce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.754ex; height:2.176ex;" alt="{\displaystyle a<x<b}"></span>)</i> oder teilweise angehören <i>(halboffenes Intervall, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq x<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq x<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd88c05ae23aff84cb261fc8b5bbf3d2a228e291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.754ex; height:2.343ex;" alt="{\displaystyle a\leq x<b}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<x\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<x\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f49fd9cfcc2576808e97594b97987bf4fb19b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.754ex; height:2.343ex;" alt="{\displaystyle a<x\leq b}"></span>).</i> </p><p>Zusammenhängend bedeutet hier: Wenn zwei Objekte in der Teilmenge enthalten sind, dann sind auch alle Objekte, die (in der Trägermenge) dazwischen liegen, darin enthalten. Die wichtigsten Beispiele für Trägermengen sind die Mengen der reellen, der <a href="/wiki/Rationale_Zahl" title="Rationale Zahl">rationalen</a>, der <a href="/wiki/Ganze_Zahl" title="Ganze Zahl">ganzen</a> und der <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a>. In den genannten Fällen und allgemeiner immer dann, wenn eine <a href="/wiki/Subtraktion" title="Subtraktion">Differenz</a> zwischen zwei Elementen der Trägermenge erklärt ist, bezeichnet man die Differenz zwischen der oberen und unteren Grenze des Intervalls (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecca61f9c918fe1deb227ed79d4979d70c443ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle b-a}"></span>) als <i>Länge</i> des Intervalls oder kurz <i>Intervalllänge;</i> für diese Differenz ist auch die Bezeichnung <i>Intervalldurchmesser</i> geläufig. Wenn ein <a href="/wiki/Arithmetisches_Mittel" title="Arithmetisches Mittel">arithmetisches Mittel</a> der Intervallgrenzen erklärt ist, wird dieses als <i>Intervallmittelpunkt</i> bezeichnet. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Beispiele"><span class="tocnumber">1</span> <span class="toctext">Beispiele</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Bezeichnungs-_und_Schreibweisen"><span class="tocnumber">2</span> <span class="toctext">Bezeichnungs- und Schreibweisen</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Beschränkte_Intervalle"><span class="tocnumber">2.1</span> <span class="toctext">Beschränkte Intervalle</span></a> <ul> <li class="toclevel-3 tocsection-4"><a href="#Abgeschlossenes_Intervall"><span class="tocnumber">2.1.1</span> <span class="toctext">Abgeschlossenes Intervall</span></a></li> <li class="toclevel-3 tocsection-5"><a href="#Offenes_Intervall"><span class="tocnumber">2.1.2</span> <span class="toctext">Offenes Intervall</span></a></li> <li class="toclevel-3 tocsection-6"><a href="#Halboffenes_(genauer_rechtsoffenes)_Intervall"><span class="tocnumber">2.1.3</span> <span class="toctext">Halboffenes (genauer rechtsoffenes) Intervall</span></a></li> <li class="toclevel-3 tocsection-7"><a href="#Halboffenes_(genauer_linksoffenes)_Intervall"><span class="tocnumber">2.1.4</span> <span class="toctext">Halboffenes (genauer linksoffenes) Intervall</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-8"><a href="#Unbeschränkte_Intervalle"><span class="tocnumber">2.2</span> <span class="toctext">Unbeschränkte Intervalle</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#n-dimensionale_Intervalle"><span class="tocnumber">3</span> <span class="toctext"><i>n</i>-dimensionale Intervalle</span></a> <ul> <li class="toclevel-2 tocsection-10"><a href="#Definition"><span class="tocnumber">3.1</span> <span class="toctext">Definition</span></a></li> <li class="toclevel-2 tocsection-11"><a href="#Beschränkte_n-dimensionale_Intervalle"><span class="tocnumber">3.2</span> <span class="toctext">Beschränkte <i>n</i>-dimensionale Intervalle</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Verallgemeinerung"><span class="tocnumber">4</span> <span class="toctext">Verallgemeinerung</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Siehe_auch"><span class="tocnumber">5</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Literatur"><span class="tocnumber">6</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Weblinks"><span class="tocnumber">7</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Einzelnachweise"><span class="tocnumber">8</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=1" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Beispiele"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>In der Menge der natürlichen Zahlen</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{5,6,7,8,9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{5,6,7,8,9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a932ed81414b5b1083f288e5724f25cffbe90c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.273ex; height:2.843ex;" alt="{\displaystyle \{5,6,7,8,9\}}"></span></dd></dl> <p>In diesem Fall einer <a href="/wiki/Diskret" title="Diskret">diskreten</a> Menge sind die Elemente des Intervalls benachbart. </p> <dl><dt>In der Menge der reellen Zahlen</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]=\{x\in \mathbb {R} \mid 0\leq x\leq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]=\{x\in \mathbb {R} \mid 0\leq x\leq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c59720e37aef81554e89f04b735e17071dc1c0ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.713ex; height:2.843ex;" alt="{\displaystyle [0,1]=\{x\in \mathbb {R} \mid 0\leq x\leq 1\}}"></span>,</dd></dl> <p>die Menge aller Zahlen zwischen 0 und 1, wobei die Endpunkte 0 und 1 mit eingeschlossen sind. </p><p>Triviale Beispiele von Intervallen sind die <a href="/wiki/Leere_Menge" title="Leere Menge">leere Menge</a> und Mengen, die genau ein Element besitzen. Wenn man diese nicht einschließen möchte, dann spricht man von <i>echten Intervallen.</i> </p><p>Die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{5,6,7,8,9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{5,6,7,8,9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a932ed81414b5b1083f288e5724f25cffbe90c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.273ex; height:2.843ex;" alt="{\displaystyle \{5,6,7,8,9\}}"></span> kann auch als Teilmenge der Trägermenge der reellen Zahlen betrachtet werden. In diesem Fall handelt es sich nicht um ein Intervall, da die Menge zum Beispiel die zwischen 6 und 7 liegenden nichtnatürlichen Zahlen nicht enthält. </p><p>Die Trägermenge der reellen Zahlen spielt insofern eine Sonderrolle unter den genannten Trägermengen für Intervalle, als sie <a href="/wiki/Ordnungsvollst%C3%A4ndig" class="mw-redirect" title="Ordnungsvollständig">ordnungsvollständig</a> ist (s. a. <a href="/wiki/Dedekindscher_Schnitt" title="Dedekindscher Schnitt">Dedekindscher Schnitt</a>). Intervalle sind in diesem Fall genau die im Sinne der <a href="/wiki/Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a> <a href="/wiki/Zusammenh%C3%A4ngender_Raum" title="Zusammenhängender Raum">zusammenhängenden</a> Teilmengen. </p> <div class="mw-heading mw-heading2"><h2 id="Bezeichnungs-_und_Schreibweisen">Bezeichnungs- und Schreibweisen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=2" title="Abschnitt bearbeiten: Bezeichnungs- und Schreibweisen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Bezeichnungs- und Schreibweisen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ein Intervall kann (beidseitig) <i>beschränkt</i> oder – auch einseitig – <i>unbeschränkt</i> sein. Es ist durch seine untere und seine obere Intervallgrenze eindeutig bestimmt, wenn zusätzlich angegeben wird, ob diese Grenzen im Intervall enthalten sind. </p><p>Es gibt zwei verschiedene häufig verwendete <i>Intervallschreibweisen:</i> </p> <ul><li>Bei der häufigeren der beiden verwendet man für Grenzen, die zum Intervall gehören, <a href="/wiki/Klammer_(Zeichen)#Eckige_Klammern" title="Klammer (Zeichen)">eckige Klammern</a> und runde für Grenzen, die nicht zum Intervall gehören. Die eckigen Klammern entsprechen einem schwachen <a href="/wiki/Vergleichszeichen" title="Vergleichszeichen">Ungleichheitszeichen</a> ≤.<sup id="cite_ref-Senger_1-0" class="reference"><a href="#cite_note-Senger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Die runden Klammern entsprechen einem starken Ungleichheitszeichen <.<sup id="cite_ref-Senger_1-1" class="reference"><a href="#cite_note-Senger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>Bei der anderen Schreibweise werden statt der runden Klammern nach außen gewendete (gespiegelte) eckige verwendet. Im Folgenden werden beide Schreibweisen gezeigt und der <i><a href="/wiki/Menge_(Mathematik)#Begriff_und_Notation_von_Mengen" title="Menge (Mathematik)">Mengenschreibweise</a></i> gegenübergestellt:</li></ul> <div class="mw-heading mw-heading3"><h3 id="Beschränkte_Intervalle"><span id="Beschr.C3.A4nkte_Intervalle"></span>Beschränkte Intervalle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=3" title="Abschnitt bearbeiten: Beschränkte Intervalle" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Beschränkte Intervalle"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>. Ein <i>beschränktes</i> Intervall mit der unteren Grenze <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und der oberen Grenze <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> ist <i><a href="/wiki/Abgeschlossene_Menge" title="Abgeschlossene Menge">abgeschlossen</a>,</i> wenn es beide Grenzen<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> enthält, und <i><a href="/wiki/Offene_Menge" title="Offene Menge">offen</a>,</i> wenn beide Grenzen nicht enthalten sind. Ein <i>beschränktes</i> Intervall heißt <i>halboffen,</i> wenn es genau eine der beiden Intervallgrenzen enthält. </p> <div class="mw-heading mw-heading4"><h4 id="Abgeschlossenes_Intervall">Abgeschlossenes Intervall</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=4" title="Abschnitt bearbeiten: Abgeschlossenes Intervall" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Abgeschlossenes Intervall"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]:=\{x\in \mathbb {R} \mid a\leq x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]:=\{x\in \mathbb {R} \mid a\leq x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/602ac1258fbfd33508ed394aaa45b93bf4f91cf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.165ex; height:2.843ex;" alt="{\displaystyle [a,b]:=\{x\in \mathbb {R} \mid a\leq x\leq b\}}"></span></dd></dl> <p>Das Intervall enthält im Normalfall (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\geq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≥</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\geq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2db6581643a6d21f1efd54d284cfcc99b850531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle b\geq a}"></span>) sowohl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> als auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Im Sonderfall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec536027666ac731c00e6db3a3b0b10d7ebbe031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b<a}"></span> ist das Intervall leer. </p><p>Ein Intervall ist genau dann <a href="/wiki/Kompaktheit_(reelle_Zahlen)" title="Kompaktheit (reelle Zahlen)"><i>kompakt</i></a>, wenn es abgeschlossen und beschränkt ist. </p> <div class="mw-heading mw-heading4"><h4 id="Offenes_Intervall">Offenes Intervall</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=5" title="Abschnitt bearbeiten: Offenes Intervall" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Offenes Intervall"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)={]a,b[}:=\{x\in \mathbb {R} \mid a<x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)={]a,b[}:=\{x\in \mathbb {R} \mid a<x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d09b46c9ddf51c14f8f7e9c54705ca44f5dfb92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.334ex; height:2.843ex;" alt="{\displaystyle (a,b)={]a,b[}:=\{x\in \mathbb {R} \mid a<x<b\}}"></span></dd></dl> <p>Das Intervall enthält weder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> noch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Die Notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> ist die traditionell verwendete, während <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {]a,b[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {]a,b[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db32da0a1eb3aab2b111c28c26604501b45ca8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle {]a,b[}}"></span> auf <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> zurückgeht.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Halboffenes_(genauer_rechtsoffenes)_Intervall"><span id="Halboffenes_.28genauer_rechtsoffenes.29_Intervall"></span>Halboffenes (genauer rechtsoffenes) Intervall</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=6" title="Abschnitt bearbeiten: Halboffenes (genauer rechtsoffenes) Intervall" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Halboffenes (genauer rechtsoffenes) Intervall"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b)={[a,b[}:=\{x\in \mathbb {R} \mid a\leq x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b)={[a,b[}:=\{x\in \mathbb {R} \mid a\leq x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8859e15f34993c8ff412f724b5d4c0923bfce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.076ex; height:2.843ex;" alt="{\displaystyle [a,b)={[a,b[}:=\{x\in \mathbb {R} \mid a\leq x<b\}}"></span></dd></dl> <p>Das Intervall enthält im Normalfall (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1cfc86ca957eea4f09d683db2412a173f6f404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b>a}"></span>) zwar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, aber nicht <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Halboffenes_(genauer_linksoffenes)_Intervall"><span id="Halboffenes_.28genauer_linksoffenes.29_Intervall"></span>Halboffenes (genauer linksoffenes) Intervall</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=7" title="Abschnitt bearbeiten: Halboffenes (genauer linksoffenes) Intervall" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Halboffenes (genauer linksoffenes) Intervall"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b]={]a,b]}:=\{x\in \mathbb {R} \mid a<x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b]={]a,b]}:=\{x\in \mathbb {R} \mid a<x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60402015ad73e30ce797a3c69c16208d93809df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.076ex; height:2.843ex;" alt="{\displaystyle (a,b]={]a,b]}:=\{x\in \mathbb {R} \mid a<x\leq b\}}"></span></dd></dl> <p>Das Intervall enthält im Normalfall (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1cfc86ca957eea4f09d683db2412a173f6f404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b>a}"></span>) zwar nicht <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, wohl aber <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. </p><p>Im Fall von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f55bc77dec8088791b5c1ed51e634cc1b431fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=1}"></span> heißt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> das <b>offene Einheitsintervall</b> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> das <b>abgeschlossene Einheitsintervall</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Unbeschränkte_Intervalle"><span id="Unbeschr.C3.A4nkte_Intervalle"></span>Unbeschränkte Intervalle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=8" title="Abschnitt bearbeiten: Unbeschränkte Intervalle" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Unbeschränkte Intervalle"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wenn auf einer Seite die Intervallgrenze fehlt, es dort also keine Schranke geben soll, spricht man von einem (auf dieser Seite) unbeschränkten Intervall. Meist werden hierfür die bekannten Symbole <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span> als „Ersatz“-Intervallgrenzen verwendet, die selbst nie<sup id="cite_ref-erw_4-0" class="reference"><a href="#cite_note-erw-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> zum Intervall gehören (deshalb die Schreibung mit runder Klammer). In mancher Literatur werden beschränkte Intervalle auch als <i>eigentlich,</i> unbeschränkte als <i>uneigentlich</i> bezeichnet. </p> <dl><dt>Linksseitig unendliches abgeschlossenes Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,b]={]{-\infty ,b}]}:=\{x\in \mathbb {R} \mid x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,b]={]{-\infty ,b}]}:=\{x\in \mathbb {R} \mid x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66acfdf806380321d3a865eaf94c861c055166db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.552ex; height:2.843ex;" alt="{\displaystyle (-\infty ,b]={]{-\infty ,b}]}:=\{x\in \mathbb {R} \mid x\leq b\}}"></span></dd></dl> <p>Es enthält alle Zahlen, die kleiner oder gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> sind. </p> <dl><dt>Linksseitig unendliches offenes Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,b)={]{-\infty ,b}[}:=\{x\in \mathbb {R} \mid x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,b)={]{-\infty ,b}[}:=\{x\in \mathbb {R} \mid x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30616ceebc6e515924f437a9c2925d6bdcc306f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.81ex; height:2.843ex;" alt="{\displaystyle (-\infty ,b)={]{-\infty ,b}[}:=\{x\in \mathbb {R} \mid x<b\}}"></span></dd></dl> <p>Es enthält alle Zahlen, die kleiner als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> sind. </p> <dl><dt>Rechtsseitig unendliches abgeschlossenes Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,\infty )={[{a,\infty }[}:=\{x\in \mathbb {R} \mid a\leq x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,\infty )={[{a,\infty }[}:=\{x\in \mathbb {R} \mid a\leq x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa93da931b52402ca6719d00cef166ed2200c1f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.632ex; height:2.843ex;" alt="{\displaystyle [a,\infty )={[{a,\infty }[}:=\{x\in \mathbb {R} \mid a\leq x\}}"></span></dd></dl> <p>Es enthält alle Zahlen, die größer oder gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> sind. </p> <dl><dt>Rechtsseitig unendliches offenes Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,\infty )={]{a,\infty }[}:=\{x\in \mathbb {R} \mid a<x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,\infty )={]{a,\infty }[}:=\{x\in \mathbb {R} \mid a<x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d646bf86a5cf1b1b5caf9974f246b7be8b84c70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.89ex; height:2.843ex;" alt="{\displaystyle (a,\infty )={]{a,\infty }[}:=\{x\in \mathbb {R} \mid a<x\}}"></span></dd></dl> <p>Es enthält alle Zahlen, die größer als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> sind. </p> <dl><dt>Beidseitig unendliches offenes (und zugleich abgeschlossenes) Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,\infty )={]{-\infty ,\infty }[}:=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,\infty )={]{-\infty ,\infty }[}:=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35dc86aacaaea8659a0f07fc9c1ae6200f0d932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.604ex; height:2.843ex;" alt="{\displaystyle (-\infty ,\infty )={]{-\infty ,\infty }[}:=\mathbb {R} }"></span></dd></dl> <p>Es enthält alle Zahlen zwischen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span>. Dies entspricht der gesamten Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> der reellen Zahlen.<sup id="cite_ref-erw_4-1" class="reference"><a href="#cite_note-erw-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Bei obiger Definition wird übrigens nicht <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></span> gefordert, sodass für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span> jedes Intervall leer ist. Daneben existieren auch je nach Anwendung Definitionen, die solche Intervalle nicht erlauben oder im Falle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span> einfach die Grenzen vertauschen. </p><p>Zur Vermeidung von Verwechslungen mit dem <a href="/wiki/Dezimalkomma" class="mw-redirect" title="Dezimalkomma">Dezimalkomma</a> wird als Trennzeichen auch das <a href="/wiki/Semikolon" title="Semikolon">Semikolon</a> (;), selten auch ein senkrechter Strich (|) verwendet, z. B. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,2{,}5]=(0;2{,}5]=(0|2{,}5].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>;</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,2{,}5]=(0;2{,}5]=(0|2{,}5].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40896aeece549b21b60cfe36f03d74ae0aac59ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.616ex; height:2.843ex;" alt="{\displaystyle (0,2{,}5]=(0;2{,}5]=(0|2{,}5].}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="n-dimensionale_Intervalle"><span id="n-dimensionales_Intervall"></span><i>n</i>-dimensionale Intervalle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=9" title="Abschnitt bearbeiten: n-dimensionale Intervalle" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: n-dimensionale Intervalle"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=10" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analog definiert man für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> im <i>n</i>-dimensionalen Raum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> ein beliebiges <b><i>n</i>-dimensionales Intervall</b> (<a href="/wiki/Quader" title="Quader">Quader</a>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I:=I_{1}\times \dotsb \times I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I:=I_{1}\times \dotsb \times I_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cd1ef5fc4766c2dbaf052f414b0c7e696e8dfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.64ex; height:2.509ex;" alt="{\displaystyle I:=I_{1}\times \dotsb \times I_{n}}"></span> mit beliebigen Intervallen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1},\dotsc ,I_{n}\subseteq \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1},\dotsc ,I_{n}\subseteq \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1538194ee8f3e93adadc71d819ec52efe289e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.92ex; height:2.509ex;" alt="{\displaystyle I_{1},\dotsc ,I_{n}\subseteq \mathbb {R} .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Beschränkte_n-dimensionale_Intervalle"><span id="Beschr.C3.A4nkte_n-dimensionale_Intervalle"></span>Beschränkte <i>n</i>-dimensionale Intervalle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=11" title="Abschnitt bearbeiten: Beschränkte n-dimensionale Intervalle" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Beschränkte n-dimensionale Intervalle"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es seien nun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7046a24cb5e4693f79379e5a6ce2de6ad3a601f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.998ex; height:2.676ex;" alt="{\displaystyle a,b\in \mathbb {R} ^{n}}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{1},\dotsc ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(a_{1},\dotsc ,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5767a80248417fc37e45eeb9a08045f7e89b3f4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.048ex; height:2.843ex;" alt="{\displaystyle a=(a_{1},\dotsc ,a_{n})}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=(b_{1},\dotsc ,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=(b_{1},\dotsc ,b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b06e0e470cd5281c1afa03234c82432d834604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.351ex; height:2.843ex;" alt="{\displaystyle b=(b_{1},\dotsc ,b_{n})}"></span>, dann gilt speziell: </p> <dl><dt>Abgeschlossenes Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}\leq b_{1},\dotsc ,a_{n}\leq x_{n}\leq b_{n}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}\leq b_{1},\dotsc ,a_{n}\leq x_{n}\leq b_{n}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4740902ff50262b094c4cfc3be1014f8db5bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.37ex; height:2.843ex;" alt="{\displaystyle [a,b]:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}\leq b_{1},\dotsc ,a_{n}\leq x_{n}\leq b_{n}\}.}"></span></dd> <dt>Offenes Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)={]{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}<b_{1},\dotsc ,a_{n}<x_{n}<b_{n}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)={]{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}<b_{1},\dotsc ,a_{n}<x_{n}<b_{n}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c20b557e648ce42b0ce302e9a10209f4429cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.539ex; height:2.843ex;" alt="{\displaystyle (a,b)={]{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}<b_{1},\dotsc ,a_{n}<x_{n}<b_{n}\}.}"></span></dd> <dt>Halboffenes (genauer rechtsoffenes) Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b)={[{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}<b_{1},\dotsc ,a_{n}\leq x_{n}<b_{n}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b)={[{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}<b_{1},\dotsc ,a_{n}\leq x_{n}<b_{n}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8306f67990be01c776f18a0220bc4258fa29831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.282ex; height:2.843ex;" alt="{\displaystyle [a,b)={[{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}<b_{1},\dotsc ,a_{n}\leq x_{n}<b_{n}\}.}"></span></dd> <dt>Halboffenes (genauer linksoffenes) Intervall</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b]={]{a,b}]}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}\leq b_{1},\dotsc ,a_{n}<x_{n}\leq b_{n}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b]={]{a,b}]}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}\leq b_{1},\dotsc ,a_{n}<x_{n}\leq b_{n}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc52b97fa6ce2450ad13bd9d6de80e4dd55f214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.282ex; height:2.843ex;" alt="{\displaystyle (a,b]={]{a,b}]}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}\leq b_{1},\dotsc ,a_{n}<x_{n}\leq b_{n}\}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung">Verallgemeinerung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=12" title="Abschnitt bearbeiten: Verallgemeinerung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a> sind reelle Intervalle Beispiele für <a href="/wiki/Zusammenh%C3%A4ngend" class="mw-redirect" title="Zusammenhängend">zusammenhängende</a> Mengen, tatsächlich ist eine Teilmenge der reellen Zahlen sogar genau dann zusammenhängend, wenn sie ein Intervall ist. Offene Intervalle sind offene Mengen und abgeschlossene Intervalle sind abgeschlossene Mengen. Halboffene Intervalle sind weder offen noch abgeschlossen. Abgeschlossene beschränkte Intervalle sind <a href="/wiki/Kompaktheit_(reelle_Zahlen)" title="Kompaktheit (reelle Zahlen)">kompakt</a>. </p><p>Alle hier für die <a href="/wiki/Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> gemachten Schreibweisen lassen sich direkt auf beliebige <a href="/wiki/Ordnungsrelationen" class="mw-redirect" title="Ordnungsrelationen">total geordnete</a> Mengen übertragen. </p><p>Bei der Übertragung auf <a href="/wiki/Halbordnung" class="mw-redirect" title="Halbordnung">Halbordnungen</a><sup id="cite_ref-nLab_interval_5-0" class="reference"><a href="#cite_note-nLab_interval-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> ist zu beachten, dass wegen fehlender Totalität Intervalle „oft“ leer sind. </p><p>Bei der Übertragung auf <a href="/wiki/Quasiordnung" title="Quasiordnung">Quasiordnungen</a> ist zu beachten, dass derartig definierte „Intervalle“ gewöhnlich „viel mehr“ Elemente enthalten. Beispielsweise bekommt man mit der für <a href="/wiki/Komplexe_Zahlen" class="mw-redirect" title="Komplexe Zahlen">komplexe Zahlen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084d8fe36677ad2f908acc909b6ff59d9e9964dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {C} }"></span> über den <a href="/wiki/Absoluter_Betrag" class="mw-redirect" title="Absoluter Betrag">Absolutbetrag</a> per „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b,{\text{ falls }}|a|\leq |b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> falls </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b,{\text{ falls }}|a|\leq |b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197b41ce22c69a33e0898436c9497162dfdcb360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.519ex; height:2.843ex;" alt="{\displaystyle a\leq b,{\text{ falls }}|a|\leq |b|}"></span>“ festgelegten Quasiordnung im Normalfall Kreisscheiben in der komplexen Zahlenebene. Eine analoge Definition im Fall eines <a href="/wiki/Metrischer_Vektorraum" class="mw-redirect" title="Metrischer Vektorraum">metrischen</a> oder allgemeiner <a href="/wiki/Normierter_Vektorraum" class="mw-redirect" title="Normierter Vektorraum">normierten Vektorraums</a> ergeben im Normalfall Kugelschalen o. ä. </p> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=13" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Intervallarithmetik" title="Intervallarithmetik">Intervallarithmetik</a></li> <li><a href="/wiki/Intervallschachtelung" title="Intervallschachtelung">Intervallschachtelung</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=14" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Harro Heuser: <i>Lehrbuch der Analysis</i>. Teil 1. 5. Auflage. Teubner-Verlag, 1988, <a href="/wiki/Spezial:ISBN-Suche/3519422212" class="internal mw-magiclink-isbn">ISBN 3-519-42221-2</a>, S. 84</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=15" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div 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="Wikibooks"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></div><b><a href="https://de.wikibooks.org/wiki/Mathe_f%C3%BCr_Nicht-Freaks:_Intervall" class="extiw" title="b:Mathe für Nicht-Freaks: Intervall">Wikibooks: Mathe für Nicht-Freaks: Intervall</a></b> – Lern- und Lehrmaterialien</div> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Intervall_(Mathematik)&veaction=edit&section=16" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Intervall_(Mathematik)&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-Senger-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Senger_1-0">a</a></sup> <sup><a href="#cite_ref-Senger_1-1">b</a></sup></span> <span class="reference-text">Jürgen Senger: <cite style="font-style:italic">Mathematik: Grundlagen für Ökonomen</cite>. Walter de Gruyter, 2009, <a href="/wiki/Spezial:ISBN-Suche/9783486710588" class="internal mw-magiclink-isbn">ISBN 978-3-486-71058-8</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>65</span> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=NyvpBQAAQBAJ&pg=PA65#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<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:Intervall+%28Mathematik%29&rft.au=J%C3%BCrgen+Senger&rft.btitle=Mathematik%3A+Grundlagen+f%C3%BCr+%C3%96konomen&rft.date=2009&rft.genre=book&rft.isbn=9783486710588&rft.pages=65&rft.pub=Walter+de+Gruyter" style="display:none"> </span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">topologisch gesehen: seinen <a href="/wiki/Rand_(Topologie)" title="Rand (Topologie)">Rand</a>, der hier aus dem linken und dem rechten Randpunkt besteht</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://hsm.stackexchange.com/questions/142/why-is-american-and-french-notation-different-for-open-intervals-x-y-vs-x/193#193"><i>Why is American and French notation different for open intervals (x, y) vs. ]x, y[?</i></a> In: <i>History of Science and Mathematics.</i> 30. Oktober 2014,<span class="Abrufdatum"> abgerufen am 10. August 2024</span> (englisch, Blog).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AIntervall+%28Mathematik%29&rft.title=Why+is+American+and+French+notation+different+for+open+intervals+%28x%2C+y%29+vs.+%26%2393%3Bx%2C+y%26%2391%3B%3F&rft.description=Why+is+American+and+French+notation+different+for+open+intervals+%28x%2C+y%29+vs.+%26%2393%3Bx%2C+y%26%2391%3B%3F&rft.identifier=https%3A%2F%2Fhsm.stackexchange.com%2Fquestions%2F142%2Fwhy-is-american-and-french-notation-different-for-open-intervals-x-y-vs-x%2F193%23193&rft.date=2014-10-30&rft.language=en"> </span></span> </li> <li id="cite_note-erw-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-erw_4-0">a</a></sup> <sup><a href="#cite_ref-erw_4-1">b</a></sup></span> <span class="reference-text">Siehe dazu jedoch die abgeschlossenen Intervalle in den <a href="/wiki/Erweiterte_reelle_Zahl#Affiner_Fall" title="Erweiterte reelle Zahl">erweiterten reellen Zahlen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {R} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a252686663952856c9c47f4a3fda0ac9a8feb112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.44ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbb {R} }}.}"></span></span> </li> <li id="cite_note-nLab_interval-5"><span class="mw-cite-backlink"><a href="#cite_ref-nLab_interval_5-0">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/interval">interval</a>. Auf: nLab. Stand: 30. 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(رياضيات) – Arabisch" lang="ar" hreflang="ar" data-title="مجال فاصل (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99di_aral%C4%B1qlar" title="Ədədi aralıqlar – Aserbaidschanisch" lang="az" hreflang="az" data-title="Ədədi aralıqlar" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B4%D1%86%D1%96%D0%BD%D0%B0%D0%BA" title="Адцінак – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Адцінак" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Интервал (математика) – Bulgarisch" lang="bg" hreflang="bg" data-title="Интервал (математика)" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Bosnisch" lang="bs" hreflang="bs" data-title="Interval (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Interval_(matem%C3%A0tiques)" title="Interval (matemàtiques) – Katalanisch" lang="ca" hreflang="ca" data-title="Interval (matemàtiques)" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D9%88%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ماوە (ماتماتیک) – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="ماوە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Tschechisch" lang="cs" hreflang="cs" data-title="Interval (matematika)" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D1%83%D1%88%C4%83%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Хушăк (математика) – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Хушăк (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfwng" title="Cyfwng – Walisisch" lang="cy" hreflang="cy" data-title="Cyfwng" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Interval_(matematik)" title="Interval (matematik) – Dänisch" lang="da" hreflang="da" data-title="Interval (matematik)" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%AC%CF%83%CF%84%CE%B7%CE%BC%CE%B1_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Διάστημα (μαθηματικά) – Griechisch" lang="el" hreflang="el" data-title="Διάστημα (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Interval_(mathematics)" title="Interval (mathematics) – Englisch" lang="en" hreflang="en" data-title="Interval (mathematics)" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Intervalo_(matematiko)" title="Intervalo (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Intervalo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Intervalo_(matem%C3%A1tica)" title="Intervalo (matemática) – Spanisch" lang="es" hreflang="es" data-title="Intervalo (matemática)" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Intervall_(matemaatika)" title="Intervall (matemaatika) – Estnisch" lang="et" hreflang="et" data-title="Intervall (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Tarte_(matematika)" title="Tarte (matematika) – Baskisch" lang="eu" hreflang="eu" data-title="Tarte (matematika)" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%D9%87" title="بازه – Persisch" lang="fa" hreflang="fa" data-title="بازه" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4li" title="Väli – Finnisch" lang="fi" hreflang="fi" data-title="Väli" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Intervalle_(math%C3%A9matiques)" title="Intervalle (mathématiques) – Französisch" lang="fr" hreflang="fr" data-title="Intervalle (mathématiques)" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Interval_(matematiche)" title="Interval (matematiche) – Friaulisch" lang="fur" hreflang="fur" data-title="Interval (matematiche)" data-language-autonym="Furlan" data-language-local-name="Friaulisch" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Eatramh_(matamaitic)" title="Eatramh (matamaitic) – Irisch" lang="ga" hreflang="ga" data-title="Eatramh (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Intervalo_(matem%C3%A1ticas)" title="Intervalo (matemáticas) – Galicisch" lang="gl" hreflang="gl" data-title="Intervalo (matemáticas)" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%98%D7%A2_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קטע (מתמטיקה) – Hebräisch" lang="he" hreflang="he" data-title="קטע (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%B0%E0%A4%BE%E0%A4%B2_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="अंतराल (गणित) – Hindi" lang="hi" hreflang="hi" data-title="अंतराल (गणित)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Kroatisch" lang="hr" hreflang="hr" data-title="Interval (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Intervallum" title="Intervallum – Ungarisch" lang="hu" hreflang="hu" data-title="Intervallum" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%AB%D5%BB%D5%A1%D5%AF%D5%A1%D5%B5%D6%84_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Միջակայք (մաթեմատիկա) – Armenisch" lang="hy" hreflang="hy" data-title="Միջակայք (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Selang_(matematika)" title="Selang (matematika) – Indonesisch" lang="id" hreflang="id" data-title="Selang (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Intervalo_(matematiko)" title="Intervalo (matematiko) – Ido" lang="io" hreflang="io" data-title="Intervalo (matematiko)" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Bil_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Bil (stærðfræði) – Isländisch" lang="is" hreflang="is" data-title="Bil (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Intervallo_(matematica)" title="Intervallo (matematica) – Italienisch" lang="it" hreflang="it" data-title="Intervallo (matematica)" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8C%BA%E9%96%93_(%E6%95%B0%E5%AD%A6)" title="区間 (数学) – Japanisch" lang="ja" hreflang="ja" data-title="区間 (数学)" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB" title="Интервал – Kasachisch" lang="kk" hreflang="kk" data-title="Интервал" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%AC%EA%B0%84" title="구간 – Koreanisch" lang="ko" hreflang="ko" data-title="구간" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Intervallum_(mathematica)" title="Intervallum (mathematica) – Latein" lang="la" hreflang="la" data-title="Intervallum (mathematica)" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Intervalas_(matematika)" title="Intervalas (matematika) – Litauisch" lang="lt" hreflang="lt" data-title="Intervalas (matematika)" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Interv%C4%81ls_(matem%C4%81tika)" title="Intervāls (matemātika) – Lettisch" lang="lv" hreflang="lv" data-title="Intervāls (matemātika)" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Интервал (математика) – Mazedonisch" lang="mk" hreflang="mk" data-title="Интервал (математика)" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Interval_(wiskunde)" title="Interval (wiskunde) – Niederländisch" lang="nl" hreflang="nl" data-title="Interval (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Intervall_i_matematikk" title="Intervall i matematikk – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Intervall i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Intervall_(matematikk)" title="Intervall (matematikk) – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Intervall (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przedzia%C5%82_(matematyka)" title="Przedział (matematyka) – Polnisch" lang="pl" hreflang="pl" data-title="Przedział (matematyka)" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Intervalo_(matem%C3%A1tica)" title="Intervalo (matemática) – Portugiesisch" lang="pt" hreflang="pt" data-title="Intervalo (matemática)" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Interval_(matematic%C4%83)" title="Interval (matematică) – Rumänisch" lang="ro" hreflang="ro" data-title="Interval (matematică)" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BC%D0%B5%D0%B6%D1%83%D1%82%D0%BE%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Промежуток (математика) – Russisch" lang="ru" hreflang="ru" data-title="Промежуток (математика)" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Interval_(mathematics)" title="Interval (mathematics) – Schottisch" lang="sco" hreflang="sco" data-title="Interval (mathematics)" data-language-autonym="Scots" data-language-local-name="Schottisch" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Serbokroatisch" lang="sh" hreflang="sh" data-title="Interval (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Interval_(mathematics)" title="Interval (mathematics) – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Interval (mathematics)" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Slowakisch" lang="sk" hreflang="sk" data-title="Interval (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Slowenisch" lang="sl" hreflang="sl" data-title="Interval (matematika)" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Intervali_(matematik%C3%AB)" title="Intervali (matematikë) – Albanisch" lang="sq" hreflang="sq" data-title="Intervali (matematikë)" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Интервал (математика) – Serbisch" lang="sr" hreflang="sr" data-title="Интервал (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – Sundanesisch" lang="su" hreflang="su" data-title="Interval (matematika)" data-language-autonym="Sunda" data-language-local-name="Sundanesisch" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Intervall_(matematik)" title="Intervall (matematik) – Schwedisch" lang="sv" hreflang="sv" data-title="Intervall (matematik)" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%9F%E0%AF%88%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="இடைவெளி (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="இடைவெளி (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%8A%E0%B9%88%E0%B8%A7%E0%B8%87_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ช่วง (คณิตศาสตร์) – Thailändisch" lang="th" hreflang="th" data-title="ช่วง (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BC%D1%96%D0%B6%D0%BE%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Проміжок (математика) – Ukrainisch" lang="uk" hreflang="uk" data-title="Проміжок (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%88%D9%82%D9%81%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="وقفہ (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="وقفہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Interval" title="Interval – Usbekisch" lang="uz" hreflang="uz" data-title="Interval" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kho%E1%BA%A3ng_(to%C3%A1n_h%E1%BB%8Dc)" title="Khoảng (toán học) – Vietnamesisch" lang="vi" hreflang="vi" data-title="Khoảng (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8C%BA%E9%97%B4" title="区间 – Wu" lang="wuu" hreflang="wuu" data-title="区间" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%97%D0%B0%D0%B2%D1%81%D0%B0%D1%80_(%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Завсар (Математика) – Kalmückisch" lang="xal" hreflang="xal" data-title="Завсар (Математика)" data-language-autonym="Хальмг" data-language-local-name="Kalmückisch" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%80%E9%96%93" title="區間 – Chinesisch" lang="zh" hreflang="zh" data-title="區間" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8D%80%E9%96%93" title="區間 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="區間" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%96%93%E8%B7%9D" title="間距 – Kantonesisch" lang="yue" hreflang="yue" data-title="間距" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q185148#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen 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