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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example_of_a_subspace_of_a_linearly_ordered_space_whose_topology_is_not_an_order_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Example of a subspace of a linearly ordered space whose topology is not an order topology</span> </div> </a> <ul id="toc-Example_of_a_subspace_of_a_linearly_ordered_space_whose_topology_is_not_an_order_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Left_and_right_order_topologies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Left_and_right_order_topologies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Left and right order topologies</span> </div> </a> <ul id="toc-Left_and_right_order_topologies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordinal_space" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ordinal_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ordinal space</span> </div> </a> <ul id="toc-Ordinal_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology_and_ordinals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topology_and_ordinals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Topology and ordinals</span> </div> </a> <button aria-controls="toc-Topology_and_ordinals-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Topology and ordinals subsection</span> </button> <ul id="toc-Topology_and_ordinals-sublist" class="vector-toc-list"> <li id="toc-Ordinals_as_topological_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordinals_as_topological_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Ordinals as topological spaces</span> </div> </a> <ul id="toc-Ordinals_as_topological_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordinal-indexed_sequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordinal-indexed_sequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Ordinal-indexed sequences</span> </div> </a> <ul id="toc-Ordinal-indexed_sequences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Certain topology in mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology (functional analysis)</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>order topology</b> is a specific <a href="/wiki/Topological_space" title="Topological space">topology</a> that can be defined on any <a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">totally ordered set</a>. It is a natural generalization of the topology of the <a href="/wiki/Real_number" title="Real number">real numbers</a> to arbitrary totally ordered sets. </p><p>If <i>X</i> is a totally ordered set, the <b>order topology</b> on <i>X</i> is generated by the <a href="/wiki/Subbase" title="Subbase">subbase</a> of "open rays" </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 \{x\mid a<x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\mid a<x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c98a1ee76be86f664b5578e0826d8fdd3953ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.25ex; height:2.843ex;" alt="{\displaystyle \{x\mid a<x\}}"></span></dd> <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 \{x\mid x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</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 \{x\mid x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c22b7fa9bf17249d450889fee8db5ebddba7903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.017ex; height:2.843ex;" alt="{\displaystyle \{x\mid x<b\}}"></span></dd></dl> <p>for all <i>a, b</i> in <i>X</i>. Provided <i>X</i> has at least two elements, this is equivalent to saying that the open <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</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 (a,b)=\{x\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> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <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\mid a<x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb6e963c2116b163e4e0d51641910e6ef350e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.515ex; height:2.843ex;" alt="{\displaystyle (a,b)=\{x\mid a<x<b\}}"></span></dd></dl> <p>together with the above rays form a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a> for the order topology. The <a href="/wiki/Open_set" title="Open set">open sets</a> in <i>X</i> are the sets that are a <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of (possibly infinitely many) such open intervals and rays. </p><p>A <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> is called <b>orderable</b> or <b>linearly orderable</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> if there exists a total order on its elements such that the order topology induced by that order and the given topology on <i>X</i> coincide. The order topology makes <i>X</i> into a <a href="/wiki/Completely_normal_space" class="mw-redirect" title="Completely normal space">completely normal</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a>. </p><p>The standard topologies on <b>R</b>, <b>Q</b>, <b>Z</b>, and <b>N</b> are the order topologies. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Induced_order_topology">Induced order topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=1" title="Edit section: Induced order topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>Y</i> is a subset of <i>X</i>, <i>X</i> a totally ordered set, then <i>Y</i> inherits a total order from <i>X</i>. The set <i>Y</i> therefore has an order topology, the <b>induced order topology</b>. As a subset of <i>X</i>, <i>Y</i> also has a <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>. The subspace topology is always at least as <a href="/wiki/Finer_topology" class="mw-redirect" title="Finer topology">fine</a> as the induced order topology, but they are not in general the same. </p><p>For example, consider the subset <i>Y</i> = {−1} ∪ {1/<i>n</i> }<sub><i>n</i>∈<b>N</b></sub> of the <a href="/wiki/Rational_number" title="Rational number">rationals</a>. Under the subspace topology, the <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> {−1} is open in <i>Y</i>, but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space. </p> <div class="mw-heading mw-heading2"><h2 id="Example_of_a_subspace_of_a_linearly_ordered_space_whose_topology_is_not_an_order_topology">Example of a subspace of a linearly ordered space whose topology is not an order topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=2" title="Edit section: Example of a subspace of a linearly ordered space whose topology is not an order topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Though the subspace topology of <i>Y</i> = {−1} ∪ {1/<i>n</i> }<sub><i>n</i>∈<b>N</b></sub> in the section above is shown not to be generated by the induced order on <i>Y</i>, it is nonetheless an order topology on <i>Y</i>; indeed, in the subspace topology every point is isolated (i.e., singleton {<i>y</i>} is open in <i>Y</i> for every <i>y</i> in <i>Y</i>), so the subspace topology is the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a> on <i>Y</i> (the topology in which every subset of <i>Y</i> is open), and the discrete topology on any set is an order topology. To define a total order on <i>Y</i> that generates the discrete topology on <i>Y</i>, simply modify the induced order on <i>Y</i> by defining −1 to be the greatest element of <i>Y</i> and otherwise keeping the same order for the other points, so that in this new order (call it say <<sub>1</sub>) we have 1/<i>n</i> <<sub>1</sub> −1 for all <i>n</i> ∈ <b>N</b>. Then, in the order topology on <i>Y</i> generated by <<sub>1</sub>, every point of <i>Y</i> is isolated in <i>Y</i>. </p><p>We wish to define here a subset <i>Z</i> of a linearly ordered topological space <i>X</i> such that no total order on <i>Z</i> generates the subspace topology on <i>Z</i>, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology. </p><p>Let <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 Z=\{-1\}\cup (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\{-1\}\cup (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4695a5bf4a411786d4f680f613f77854dd677cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.825ex; height:2.843ex;" alt="{\displaystyle Z=\{-1\}\cup (0,1)}"></span> in the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>. The same argument as before shows that the subspace topology on <i>Z</i> is not equal to the induced order topology on <i>Z</i>, but one can show that the subspace topology on <i>Z</i> cannot be equal to any order topology on <i>Z</i>. </p><p>An argument follows. Suppose by way of contradiction that there is some <a href="/wiki/Totally_ordered_set#Strict_total_order" class="mw-redirect" title="Totally ordered set">strict total order</a> < on <i>Z</i> such that the order topology generated by < is equal to the subspace topology on <i>Z</i> (note that we are not assuming that < is the induced order on <i>Z</i>, but rather an arbitrarily given total order on <i>Z</i> that generates the subspace topology). </p><p>Let <i>M</i> = <i>Z</i> \ {−1} = (0,1), then <i>M</i> is <a href="/wiki/Connected_space" title="Connected space">connected</a>, so <i>M</i> is dense on itself and has no gaps, in regards to <. If −1 is not the smallest or the largest element of <i>Z</i>, then <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 ,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eecbbaf03fa846f2e0c57bf5f65e760d2f7e484d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.946ex; height:2.843ex;" alt="{\displaystyle (-\infty ,-1)}"></span> and <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 (-1,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec84bcdac123f01f2ff7ffc65e4feb642b6b576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:2.843ex;" alt="{\displaystyle (-1,\infty )}"></span> separate <i>M</i>, a contradiction. Assume without loss of generality that −1 is the smallest element of <i>Z</i>. Since {−1} is open in <i>Z</i>, there is some point <i>p</i> in <i>M</i> such that the interval (−1,<i>p</i>) is <a href="/wiki/Empty_set" title="Empty set">empty</a>, so <i>p</i> is the minimum of <i>M</i>. Then <i>M</i> \ {<i>p</i>} = (0,<i>p</i>) ∪ (<i>p</i>,1) is not connected with respect to the subspace topology inherited from <span class="texhtml"><b>R</b></span>. On the other hand, the subspace topology of <i>M</i> \ {<i>p</i>} inherited from the order topology of <i>Z</i> coincides with the order topology of <i>M</i> \ {<i>p</i>} induced by <, which is connected since there are no gaps in <i>M</i> \ {<i>p</i>} and it is dense. This is a contradiction. </p> <div class="mw-heading mw-heading2"><h2 id="Left_and_right_order_topologies">Left and right order topologies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=3" title="Edit section: Left and right order topologies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several variants of the order topology can be given: </p> <ul><li>The <b>right order topology</b><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> on <i>X</i> is the topology having as a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a> all intervals of the form <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 )=\{x\in X\mid x>a\}}"> <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> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>∣</mo> <mi>x</mi> <mo>></mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,\infty )=\{x\in X\mid x>a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696c27afcaa83587a26c11bdde62207e7f846773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.566ex; height:2.843ex;" alt="{\displaystyle (a,\infty )=\{x\in X\mid x>a\}}"></span>, together with the set <i>X</i>.</li> <li>The <b>left order topology</b> on <i>X</i> is the topology having as a base all intervals of the form <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 ,a)=\{x\in X\mid x<a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>∣</mo> <mi>x</mi> <mo><</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,a)=\{x\in X\mid x<a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a304321afa82143ca7832556044fbf3a54a508c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.374ex; height:2.843ex;" alt="{\displaystyle (-\infty ,a)=\{x\in X\mid x<a\}}"></span>, together with the set <i>X</i>.</li></ul> <p>The left and right order topologies can be used to give <a href="/wiki/Counterexample" title="Counterexample">counterexamples</a> in general topology. For example, the left or right order topology on a bounded set provides an example of a <a href="/wiki/Compact_space" title="Compact space">compact space</a> that is not Hausdorff. </p><p>The left order topology is the standard topology used for many <a href="/wiki/Set-theoretic" class="mw-redirect" title="Set-theoretic">set-theoretic</a> purposes on a <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (April 2021)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ordinal_space">Ordinal space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=4" title="Edit section: Ordinal space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> <i>λ</i> one can consider the spaces of ordinal numbers </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,\lambda )=\{\alpha \mid \alpha <\lambda \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mo>∣</mo> <mi>α</mi> <mo><</mo> <mi>λ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\lambda )=\{\alpha \mid \alpha <\lambda \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5debcdd74113dcce1d76dff31c1c743164bd2fe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.893ex; height:2.843ex;" alt="{\displaystyle [0,\lambda )=\{\alpha \mid \alpha <\lambda \}}"></span></dd> <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,\lambda ]=\{\alpha \mid \alpha \leq \lambda \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mo>∣</mo> <mi>α</mi> <mo>≤</mo> <mi>λ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\lambda ]=\{\alpha \mid \alpha \leq \lambda \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e469e57aa87041df20c6275a3b21c0795321ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.635ex; height:2.843ex;" alt="{\displaystyle [0,\lambda ]=\{\alpha \mid \alpha \leq \lambda \}}"></span></dd></dl> <p>together with the natural order topology. These spaces are called <b>ordinal spaces</b>. (Note that in the usual set-theoretic construction of ordinal numbers we have <i>λ</i> = [0, <i>λ</i>) and <i>λ</i> + 1 = [0, <i>λ</i>]). Obviously, these spaces are mostly of interest when <i>λ</i> is an infinite ordinal; for finite ordinals, the order topology is simply the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a>. </p><p>When <i>λ</i> = ω (the first infinite ordinal), the space [0,ω) is just <b>N</b> with the usual (still discrete) topology, while [0,ω] is the <a href="/wiki/Alexandroff_extension" title="Alexandroff extension">one-point compactification</a> of <b>N</b>. </p><p>Of particular interest is the case when <i>λ</i> = ω<sub>1</sub>, the set of all countable ordinals, and the <a href="/wiki/First_uncountable_ordinal" title="First uncountable ordinal">first uncountable ordinal</a>. The element ω<sub>1</sub> is a <a href="/wiki/Limit_point" class="mw-redirect" title="Limit point">limit point</a> of the subset [0,ω<sub>1</sub>) even though no <a href="/wiki/Sequence" title="Sequence">sequence</a> of elements in [0,ω<sub>1</sub>) has the element ω<sub>1</sub> as its limit. In particular, [0,ω<sub>1</sub>] is not <a href="/wiki/First-countable_space" title="First-countable space">first-countable</a>. The subspace [0,ω<sub>1</sub>) is first-countable however, since the only point in [0,ω<sub>1</sub>] without a countable <a href="/wiki/Local_base" class="mw-redirect" title="Local base">local base</a> is ω<sub>1</sub>. Some further properties include </p> <ul><li>neither [0,ω<sub>1</sub>) or [0,ω<sub>1</sub>] is <a href="/wiki/Separable_space" title="Separable space">separable</a> or <a href="/wiki/Second-countable" class="mw-redirect" title="Second-countable">second-countable</a></li> <li>[0,ω<sub>1</sub>] is <a href="/wiki/Compact_space" title="Compact space">compact</a>, while [0,ω<sub>1</sub>) is <a href="/wiki/Sequentially_compact_space" title="Sequentially compact space">sequentially compact</a> and <a href="/wiki/Countably_compact_space" title="Countably compact space">countably compact</a>, but not compact or <a href="/wiki/Paracompact" class="mw-redirect" title="Paracompact">paracompact</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Topology_and_ordinals">Topology and ordinals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=5" title="Edit section: Topology and ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ordinals_as_topological_spaces">Ordinals as topological spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=6" title="Edit section: Ordinals as topological spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are <a href="/wiki/Well-order" title="Well-order">well-ordered</a>, so in particular <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a>). Unless otherwise specified, this is the usual topology given to ordinals. Moreover, if we are willing to accept a <a href="/wiki/Proper_class" class="mw-redirect" title="Proper class">proper class</a> as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology. </p><p>The set of <a href="/wiki/Limit_point" class="mw-redirect" title="Limit point">limit points</a> of an ordinal <i>α</i> is precisely the set of <a href="/wiki/Limit_ordinal" title="Limit ordinal">limit ordinals</a> less than <i>α</i>. <a href="/wiki/Successor_ordinal" title="Successor ordinal">Successor ordinals</a> (and zero) less than <i>α</i> are <a href="/wiki/Isolated_point" title="Isolated point">isolated points</a> in <i>α</i>. In particular, the finite ordinals and ω are <a href="/wiki/Discrete_space" title="Discrete space">discrete</a> topological spaces, and no ordinal beyond that is discrete. The ordinal <i>α</i> is <a href="/wiki/Compact_space" title="Compact space">compact</a> as a topological space if and only if <i>α</i> is either a <a href="/wiki/Successor_ordinal" title="Successor ordinal">successor ordinal</a> or zero. </p><p>The <a href="/wiki/Closed_set" title="Closed set">closed sets</a> of a limit ordinal <i>α</i> are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it. </p><p>Any ordinal is, of course, an open subset of any larger ordinal. We can also define the topology on the ordinals in the following <a href="/wiki/Recursion" title="Recursion">inductive</a> way: 0 is the empty topological space, <i>α</i>+1 is obtained by taking the <a href="/wiki/Compactification_(mathematics)" title="Compactification (mathematics)">one-point compactification</a> of <i>α</i>, and for <i>δ</i> a limit ordinal, <i>δ</i> is equipped with the <a href="/wiki/Direct_limit" title="Direct limit">inductive limit</a> topology. Note that if <i>α</i> is a successor ordinal, then <i>α</i> is compact, in which case its one-point compactification <i>α</i>+1 is the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of <i>α</i> and a point. </p><p>As topological spaces, all the ordinals are <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> and even <a href="/wiki/Normal_space" title="Normal space">normal</a>. They are also <a href="/wiki/Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a> (connected components are points), <a href="/wiki/Scattered_space" title="Scattered space">scattered</a> (every non-empty subspace has an isolated point; in this case, just take the smallest element), <a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">zero-dimensional</a> (the topology has a <a href="/wiki/Clopen" class="mw-redirect" title="Clopen">clopen</a> <a href="/wiki/Basis_(topology)" class="mw-redirect" title="Basis (topology)">basis</a>: here, write an open interval (<i>β</i>,<i>γ</i>) as the union of the clopen intervals (<i>β</i>,<i>γ</i>'+1) = [<i>β</i>+1,<i>γ</i>'] for <i>γ</i>'<<i>γ</i>). However, they are not <a href="/wiki/Extremally_disconnected" class="mw-redirect" title="Extremally disconnected">extremally disconnected</a> in general (there are open sets, for example the even numbers from ω, whose <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closure</a> is not open). </p><p>The topological spaces ω<sub>1</sub> and its successor ω<sub>1</sub>+1 are frequently used as textbook examples of uncountable topological spaces. For example, in the topological space ω<sub>1</sub>+1, the element ω<sub>1</sub> is in the closure of the subset ω<sub>1</sub> even though no sequence of elements in ω<sub>1</sub> has the element ω<sub>1</sub> as its limit: an element in ω<sub>1</sub> is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one. </p><p>The space ω<sub>1</sub> is <a href="/wiki/First-countable_space" title="First-countable space">first-countable</a> but not <a href="/wiki/Second-countable_space" title="Second-countable space">second-countable</a>, and ω<sub>1</sub>+1 has neither of these two properties, despite being <a href="/wiki/Compact_space" title="Compact space">compact</a>. It is also worthy of note that any <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> from ω<sub>1</sub> to <b>R</b> (the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>) is eventually constant: so the <a href="/wiki/Stone%E2%80%93%C4%8Cech_compactification" title="Stone–Čech compactification">Stone–Čech compactification</a> of ω<sub>1</sub> is ω<sub>1</sub>+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much <i>larger</i> than ω). </p> <div class="mw-heading mw-heading3"><h3 id="Ordinal-indexed_sequences">Ordinal-indexed sequences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=7" title="Edit section: Ordinal-indexed sequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>α</i> is a limit ordinal and <i>X</i> is a set, an <i>α</i>-indexed sequence of elements of <i>X</i> merely means a function from <i>α</i> to <i>X</i>. This concept, a <b>transfinite sequence</b> or <b>ordinal-indexed sequence</b>, is a generalization of the concept of a <a href="/wiki/Sequence" title="Sequence">sequence</a>. An ordinary sequence corresponds to the case <i>α</i> = ω. </p><p>If <i>X</i> is a topological space, we say that an <i>α</i>-indexed sequence of elements of <i>X</i> <i>converges</i> to a limit <i>x</i> when it converges as a <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">net</a>, in other words, when given any <a href="/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> <i>U</i> of <i>x</i> there is an ordinal <i>β</i> < <i>α</i> such that <i>x</i><sub><i>ι</i></sub> is in <i>U</i> for all <i>ι</i> ≥ <i>β</i>. </p><p>Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω<sub>1</sub> is a limit point of ω<sub>1</sub>+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω<sub>1</sub>-indexed sequence which maps any ordinal less than ω<sub>1</sub> to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω<sub>1</sub>, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable. </p><p>However, ordinal-indexed sequences are not powerful enough to replace nets (or <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filters</a>) in general: for example, on the <a href="/wiki/Tychonoff_plank" title="Tychonoff plank">Tychonoff plank</a> (the product space <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 (\omega _{1}+1)\times (\omega +1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega _{1}+1)\times (\omega +1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4bdb9b368101fd9034e419f85f5f8167045d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.411ex; height:2.843ex;" alt="{\displaystyle (\omega _{1}+1)\times (\omega +1)}"></span>), the corner point <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 (\omega _{1},\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega _{1},\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4639128ab29706a8342fbd90ea1ca32604ec9f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.789ex; height:2.843ex;" alt="{\displaystyle (\omega _{1},\omega )}"></span> is a limit point (it is in the closure) of the open subset <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 \omega _{1}\times \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}\times \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14494c5f434cfc007c7e04edffd6f751b66fbc1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.786ex; height:2.009ex;" alt="{\displaystyle \omega _{1}\times \omega }"></span>, but it is not the limit of an ordinal-indexed sequence. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_topologies" title="List of topologies">List of topologies</a></li> <li><a href="/wiki/Lower_limit_topology" title="Lower limit topology">Lower limit topology</a></li> <li><a href="/wiki/Long_line_(topology)" title="Long line (topology)">Long line (topology)</a></li> <li><a href="/wiki/Linear_continuum" title="Linear continuum">Linear continuum</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology (functional analysis)</a></li> <li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLynn1962" class="citation journal cs1">Lynn, I. L. (1962). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1962-0138089-6">"Linearly orderable spaces"</a>. <i><a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a></i>. <b>13</b> (3): 454–456. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1962-0138089-6">10.1090/S0002-9939-1962-0138089-6</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Linearly+orderable+spaces&rft.volume=13&rft.issue=3&rft.pages=454-456&rft.date=1962&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1962-0138089-6&rft.aulast=Lynn&rft.aufirst=I.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9939-1962-0138089-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrder+topology" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Steen & Seebach, p. 74</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_topology&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Lynn_Arthur_Steen" class="mw-redirect" title="Lynn Arthur Steen">Steen, Lynn A.</a> and <a href="/wiki/J._Arthur_Seebach,_Jr." class="mw-redirect" title="J. Arthur Seebach, Jr.">Seebach, J. Arthur Jr.</a>; <i><a href="/wiki/Counterexamples_in_Topology" title="Counterexamples in Topology">Counterexamples in Topology</a></i>, Holt, Rinehart and Winston (1970). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-079485-4" title="Special:BookSources/0-03-079485-4">0-03-079485-4</a>.</li> <li>Stephen Willard, <i>General Topology</i>, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.</li></ul> </div> <p><i>This article incorporates material from Order topology on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Order_theory" title="Template:Order theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" title="Special:EditPage/Template:Order theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Order_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Order_theory" title="Order theory">Order theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_order_theory_topics" title="List of order theory topics">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a href="/wiki/Mirsky%27s_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<small><a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a></small>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Semiorder" title="Semiorder">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a class="mw-selflink selflink">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a></li> <li><a href="/wiki/Young%27s_lattice" title="Young's lattice">Young's lattice</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div 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