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Available in 13 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-13" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kompakt-Offen-Topologie" title="Kompakt-Offen-Topologie – German" lang="de" hreflang="de" data-title="Kompakt-Offen-Topologie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Topolog%C3%ADa_compacto-abierta" title="Topología compacto-abierta – Spanish" lang="es" hreflang="es" data-title="Topología compacto-abierta" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Topologie_compacte-ouverte" title="Topologie compacte-ouverte – French" lang="fr" hreflang="fr" data-title="Topologie compacte-ouverte" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8-%EC%97%B4%EB%A6%B0%EC%A7%91%ED%95%A9_%EC%9C%84%EC%83%81" title="콤팩트-열린집합 위상 – Korean" lang="ko" hreflang="ko" data-title="콤팩트-열린집합 위상" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94_%D7%94%D7%A7%D7%95%D7%9E%D7%A4%D7%A7%D7%98%D7%99%D7%AA-%D7%A4%D7%AA%D7%95%D7%97%D7%94" title="הטופולוגיה הקומפקטית-פתוחה – Hebrew" lang="he" hreflang="he" data-title="הטופולוגיה הקומפקטית-פתוחה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B3%E3%83%B3%E3%83%91%E3%82%AF%E3%83%88%E9%96%8B%E4%BD%8D%E7%9B%B8" title="コンパクト開位相 – Japanese" lang="ja" hreflang="ja" data-title="コンパクト開位相" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Topologia_zwarto-otwarta" title="Topologia zwarto-otwarta – Polish" lang="pl" hreflang="pl" data-title="Topologia zwarto-otwarta" data-language-autonym="Polski" data-language-local-name="Polish" 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/Topologia_compacto-aberto" title="Topologia compacto-aberto – Portuguese" lang="pt" hreflang="pt" data-title="Topologia compacto-aberto" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%BE-%D0%BE%D1%82%D0%BA%D1%80%D1%8B%D1%82%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Компактно-открытая топология – Russian" lang="ru" hreflang="ru" data-title="Компактно-открытая топология" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/T%C4%B1k%C4%B1z-a%C3%A7%C4%B1k_topoloji" title="Tıkız-açık topoloji – Turkish" lang="tr" hreflang="tr" data-title="Tıkız-açık topoloji" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%BE-%D0%B2%D1%96%D0%B4%D0%BA%D1%80%D0%B8%D1%82%D0%B0_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%8F" title="Компактно-відкрита топологія – Ukrainian" lang="uk" hreflang="uk" data-title="Компактно-відкрита топологія" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div 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">Type of topology</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>compact-open topology</b> is a <a href="/wiki/Topological_space" title="Topological space">topology</a> defined on the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Continuous_function" title="Continuous function">continuous maps</a> between two <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. The compact-open topology is one of the commonly used topologies on <a href="/wiki/Function_space" title="Function space">function spaces</a>, and is applied in <a href="/wiki/Homotopy_theory" title="Homotopy theory">homotopy theory</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. It was introduced by <a href="/wiki/Ralph_Fox" title="Ralph Fox">Ralph Fox</a> in 1945.<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> </p><p>If the <a href="/wiki/Codomain" title="Codomain">codomain</a> of the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> under consideration has a <a href="/wiki/Uniform_space" title="Uniform space">uniform structure</a> or a <a href="/wiki/Metric_space" title="Metric space">metric structure</a> then the compact-open topology is the "topology of <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a> on <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact sets</a>." That is to say, a <a href="/wiki/Sequence" title="Sequence">sequence</a> of functions <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">converges</a> in the compact-open topology precisely when it converges uniformly on every compact subset of the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact-open_topology&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">Y</span> be two <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>, and let <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span> denote the set of all <a href="/wiki/Continuous_map" class="mw-redirect" title="Continuous map">continuous maps</a> between <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">Y</span>. Given a <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact subset</a> <span class="texhtml mvar" style="font-style:italic;">K</span> of <span class="texhtml mvar" style="font-style:italic;">X</span> and an <a href="/wiki/Open_set" title="Open set">open subset</a> <span class="texhtml mvar" style="font-style:italic;">U</span> of <span class="texhtml mvar" style="font-style:italic;">Y</span>, let <span class="texhtml"><i>V</i>(<i>K</i>, <i>U</i>)</span> denote the set of all functions <span class="texhtml"> <i>f</i>  ∈ <i>C</i>(<i>X</i>, <i>Y</i>)</span> such that <span class="texhtml"> <i>f</i> (<i>K</i>) ⊆ <i>U</i>.</span> In other words, <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 V(K,U)=C(K,U)\times _{C(K,Y)}C(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">)</mo> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(K,U)=C(K,U)\times _{C(K,Y)}C(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a85e35a983b9acfd3f3f0e524e257d7d0e66b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.172ex; height:3.176ex;" alt="{\displaystyle V(K,U)=C(K,U)\times _{C(K,Y)}C(X,Y)}"></span>. Then the collection of all such <span class="texhtml"><i>V</i>(<i>K</i>, <i>U</i>)</span> is a <a href="/wiki/Subbase" title="Subbase">subbase</a> for the compact-open topology on <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span>. (This collection does not always form a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a> for a topology on <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span>.) </p><p>When working in the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of <a href="/wiki/Compactly_generated_space" title="Compactly generated space">compactly generated spaces</a>, it is common to modify this definition by restricting to the subbase formed from those <span class="texhtml mvar" style="font-style:italic;">K</span> that are the image of a <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a>. Of course, if <span class="texhtml mvar" style="font-style:italic;">X</span> is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of <a href="/wiki/Weak_Hausdorff_space" title="Weak Hausdorff space">compactly generated weak Hausdorff</a> spaces to be <a href="/wiki/Cartesian_closed_category" title="Cartesian closed category">Cartesian closed</a>, among other useful properties.<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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The confusion between this definition and the one above is caused by differing usage of the word <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact</a>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">X</span> is locally compact, 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 X\times -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c90fc45ce3aa0fb9a011d7bd2f7aa6e20e10ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.629ex; height:2.343ex;" alt="{\displaystyle X\times -}"></span> from the category of topological spaces always has a right adjoint <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 Hom(X,-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>o</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Hom(X,-)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d925f5846159d82bd06e1cdd44279f32eacc7b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.863ex; height:2.843ex;" alt="{\displaystyle Hom(X,-)}"></span>. This adjoint coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed as taking the adjoint of the product in the category of compactly generated spaces instead of the category of topological spaces, which ensures that the right adjoint always exists. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact-open_topology&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>If <span class="texhtml">*</span> is a one-point space then one can identify <span class="texhtml"><i>C</i>(*, <i>Y</i>)</span> with <span class="texhtml mvar" style="font-style:italic;">Y</span>, and under this identification the compact-open topology agrees with the topology on <span class="texhtml mvar" style="font-style:italic;">Y</span>. More generally, if <span class="texhtml mvar" style="font-style:italic;">X</span> is a <a href="/wiki/Discrete_space" title="Discrete space">discrete space</a>, then <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span> can be identified with the <a href="/wiki/Cartesian_product" title="Cartesian product">cartesian product</a> of <span class="texhtml">|<i>X</i>|</span> copies of <span class="texhtml mvar" style="font-style:italic;">Y</span> and the compact-open topology agrees with the <a href="/wiki/Product_topology" title="Product topology">product topology</a>.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">Y</span> is <span class="texhtml"><a href="/wiki/T0_space" class="mw-redirect" title="T0 space"><i>T</i><sub>0</sub></a></span>, <span class="texhtml"><a href="/wiki/T1_space" title="T1 space"><i>T</i><sub>1</sub></a></span>, <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, <a href="/wiki/Regular_space" title="Regular space">regular</a>, or <a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff</a>, then the compact-open topology has the corresponding <a href="/wiki/Separation_axiom" title="Separation axiom">separation axiom</a>.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">X</span> is Hausdorff and <span class="texhtml mvar" style="font-style:italic;">S</span> is a <a href="/wiki/Subbase" title="Subbase">subbase</a> for <span class="texhtml mvar" style="font-style:italic;">Y</span>, then the collection <span class="texhtml">{<i>V</i>(<i>K</i>, <i>U</i>) : <i>U</i> ∈ <i>S</i>, <i>K</i> compact} </span>is a <a href="/wiki/Subbase" title="Subbase">subbase</a> for the compact-open topology on <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>If <span class="texhtml mvar" style="font-style:italic;">Y</span> is a <a href="/wiki/Metric_space" title="Metric space">metric space</a> (or more generally, a <a href="/wiki/Uniform_space" title="Uniform space">uniform space</a>), then the compact-open topology is equal to the <a href="/wiki/Topology_of_compact_convergence" class="mw-redirect" title="Topology of compact convergence">topology of compact convergence</a>. In other words, if <span class="texhtml mvar" style="font-style:italic;">Y</span> is a metric space, then a sequence <span class="texhtml">{ <i>f</i><sub><i>n</i></sub> } </span><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">converges</a> to <span class="texhtml"> <i>f</i> </span> in the compact-open topology if and only if for every compact subset <span class="texhtml mvar" style="font-style:italic;">K</span> of <span class="texhtml mvar" style="font-style:italic;">X</span>, <span class="texhtml">{ <i>f</i><sub><i>n</i></sub> } </span>converges uniformly to <span class="texhtml"> <i>f</i> </span> on <span class="texhtml mvar" style="font-style:italic;">K</span>. If <span class="texhtml mvar" style="font-style:italic;">X</span> is compact and <span class="texhtml mvar" style="font-style:italic;">Y</span> is a uniform space, then the compact-open topology is equal to the topology of <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a>.</li> <li>If <span class="texhtml"><i>X</i>, <i>Y</i></span> and <span class="texhtml mvar" style="font-style:italic;">Z</span> are topological spaces, with <span class="texhtml mvar" style="font-style:italic;">Y</span> <a href="/wiki/Locally_compact_Hausdorff" class="mw-redirect" title="Locally compact Hausdorff">locally compact Hausdorff</a> (or even just locally compact <a href="/wiki/Preregular_space" class="mw-redirect" title="Preregular space">preregular</a>), then the <a href="/wiki/Function_composition" title="Function composition">composition map</a> <span class="texhtml"><i>C</i>(<i>Y</i>, <i>Z</i>) × <i>C</i>(<i>X</i>, <i>Y</i>) → <i>C</i>(<i>X</i>, <i>Z</i>),</span> given by <span class="texhtml">( <i>f</i> , <i>g</i>) ↦  <i>f</i> ∘ <i>g</i>,</span> is continuous (here all the function spaces are given the compact-open topology and <span class="texhtml"><i>C</i>(<i>Y</i>, <i>Z</i>) × <i>C</i>(<i>X</i>, <i>Y</i>)</span> is given the <a href="/wiki/Product_topology" title="Product topology">product topology</a>).</li> <li>If <span class="texhtml mvar" style="font-style:italic;">X</span> is a locally compact Hausdorff (or preregular) space, then the evaluation map <span class="texhtml"><i>e</i> : <i>C</i>(<i>X</i>, <i>Y</i>) × <i>X</i> → <i>Y</i></span>, defined by <span class="texhtml"><i>e</i>( <i>f</i> , <i>x</i>) =  <i>f</i> (<i>x</i>)</span>, is continuous. This can be seen as a special case of the above where <span class="texhtml mvar" style="font-style:italic;">X</span> is a one-point space.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">X</span> is compact, and <span class="texhtml mvar" style="font-style:italic;">Y</span> is a metric space with <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> <span class="texhtml mvar" style="font-style:italic;">d</span>, then the compact-open topology on <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span> is <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a>, and a metric for it is given by <span class="texhtml"><i>e</i>( <i>f</i> , <i>g</i>) = <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">sup</a>{<i>d</i>( <i>f</i> (<i>x</i>), <i>g</i>(<i>x</i>)) : <i>x</i> in <i>X</i>},</span> for <span class="texhtml"> <i>f</i> , <i>g</i></span> in <span class="texhtml"><i>C</i>(<i>X</i>, <i>Y</i>)</span>. More generally, if <span class="texhtml mvar" style="font-style:italic;">X</span> is <a href="/wiki/Hemicompact_space" title="Hemicompact space">hemicompact</a>, and <span class="texhtml mvar" style="font-style:italic;">Y</span> metric, the compact-open topology is metrizable by the <a href="/wiki/Hemicompact_space#Applications" title="Hemicompact space">construction linked here</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact-open_topology&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The compact open topology can be used to topologize the following sets:<sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ul><li><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 (X,x_{0})=\{f:I\to X:f(0)=f(1)=x_{0}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega (X,x_{0})=\{f:I\to X:f(0)=f(1)=x_{0}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f8d273bc2abd8382c0192b3717b51a2764a363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.309ex; height:2.843ex;" alt="{\displaystyle \Omega (X,x_{0})=\{f:I\to X:f(0)=f(1)=x_{0}\}}"></span>, the <a href="/wiki/Loop_space" title="Loop space">loop space</a> of <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> at <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>,</li> <li><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 E(X,x_{0},x_{1})=\{f:I\to X:f(0)=x_{0}{\text{ and }}f(1)=x_{1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X,x_{0},x_{1})=\{f:I\to X:f(0)=x_{0}{\text{ and }}f(1)=x_{1}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df972e59430b62e2e8dadb8075214cb262b7dbff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.117ex; height:2.843ex;" alt="{\displaystyle E(X,x_{0},x_{1})=\{f:I\to X:f(0)=x_{0}{\text{ and }}f(1)=x_{1}\}}"></span>,</li> <li><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 E(X,x_{0})=\{f:I\to X:f(0)=x_{0}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X,x_{0})=\{f:I\to X:f(0)=x_{0}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7112647e8999863ed440fab6d94f245e1523b46d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.058ex; height:2.843ex;" alt="{\displaystyle E(X,x_{0})=\{f:I\to X:f(0)=x_{0}\}}"></span>.</li></ul> <p>In addition, there is a <a href="/wiki/Homotopy#Homotopy_equivalence" title="Homotopy">homotopy equivalence</a> between the spaces <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 C(\Sigma X,Y)\cong C(X,\Omega Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\Sigma X,Y)\cong C(X,\Omega Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ecaa9d0e3b1738b69920ca9abacada47c82711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.18ex; height:2.843ex;" alt="{\displaystyle C(\Sigma X,Y)\cong C(X,\Omega Y)}"></span>.<sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> These topological spaces, <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 C(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc34e8f08e76555abf1181aae10f5cddd68fdd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.363ex; height:2.843ex;" alt="{\displaystyle C(X,Y)}"></span> are useful in homotopy theory because it can be used to form a topological space and a model for the homotopy type of the <i>set</i> of homotopy classes of maps </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 \pi (X,Y)=\{[f]:X\to Y|f{\text{ is a homotopy class}}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a homotopy class</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (X,Y)=\{[f]:X\to Y|f{\text{ is a homotopy class}}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba6352396d8bcb3b6a45787ecf8586024f205b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.664ex; height:2.843ex;" alt="{\displaystyle \pi (X,Y)=\{[f]:X\to Y|f{\text{ is a homotopy class}}\}.}"></span></dd></dl> <p>This is because <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 \pi (X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2958e49a88f446d45b6b494da1997d8b051881e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.929ex; height:2.843ex;" alt="{\displaystyle \pi (X,Y)}"></span> is the set of path components in <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 C(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc34e8f08e76555abf1181aae10f5cddd68fdd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.363ex; height:2.843ex;" alt="{\displaystyle C(X,Y)}"></span>, that is, there is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> of sets </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 \pi (X,Y)\to C(I,C(X,Y))/\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (X,Y)\to C(I,C(X,Y))/\sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff464be2a2c2d49c38c169d7ce6ad69d5ca549b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.303ex; height:2.843ex;" alt="{\displaystyle \pi (X,Y)\to C(I,C(X,Y))/\sim }"></span></dd></dl> <p>where <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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> is the homotopy equivalence. </p> <div class="mw-heading mw-heading2"><h2 id="Fréchet_differentiable_functions"><span id="Fr.C3.A9chet_differentiable_functions"></span>Fréchet differentiable functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact-open_topology&action=edit&section=4" title="Edit section: Fréchet differentiable functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">Y</span> be two <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> defined over the same <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, and let <span class="texhtml"><i>C<sup> m</sup></i>(<i>U</i>, <i>Y</i>)</span> denote the set of all <span class="texhtml mvar" style="font-style:italic;">m</span>-continuously <a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet-differentiable</a> functions from the open subset <span class="texhtml"><i>U</i> ⊆ <i>X</i></span> to <span class="texhtml mvar" style="font-style:italic;">Y</span>. The compact-open topology is the <a href="/wiki/Initial_topology" title="Initial topology">initial topology</a> induced by the <a href="/wiki/Seminorm" title="Seminorm">seminorms</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 p_{K}(f)=\sup \left\{\left\|D^{j}f(x)\right\|\ :\ x\in K,0\leq j\leq m\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>K</mi> <mo>,</mo> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>m</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{K}(f)=\sup \left\{\left\|D^{j}f(x)\right\|\ :\ x\in K,0\leq j\leq m\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a09138d9a2cf55d6d3eb314557b967616b0f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:46.427ex; height:3.509ex;" alt="{\displaystyle p_{K}(f)=\sup \left\{\left\|D^{j}f(x)\right\|\ :\ x\in K,0\leq j\leq m\right\}}"></span></dd></dl> <p>where <span class="texhtml"><i>D</i><sup>0</sup> <i>f</i> (<i>x</i>) =  <i>f</i> (<i>x</i>)</span>, for each compact subset <span class="texhtml"><i>K</i> ⊆ <i>U</i></span>.<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="Is this original research showing that this definition is equivalent in this special case to the general definition given above? Or is it a definition copied from an external reference, in which case that reference should be cited? (February 2022)">clarification needed</span></a></i>]</sup> </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=Compact-open_topology&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Topology of uniform convergence</a></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform convergence</a> – Mode of convergence of a function sequence</li></ul> <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=Compact-open_topology&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <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="CITEREFFox1945" class="citation journal cs1">Fox, Ralph H. 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