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<title>Totally disconnected space</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Totally disconnected space</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Extremally_disconnected_space" title="Extremally disconnected space">extremally disconnected space</a>.</div>

<p>In <a href="Topology" title="Topology">topology</a> and related branches of <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>totally disconnected space</b> is a <a href="Topological_space" title="Topological space">topological space</a> that has only <a href="Singleton_(mathematics)" title="Singleton (mathematics)">singletons</a> as <a href="Connected_space" title="Connected space">connected</a> <a href="Subset" title="Subset">subsets</a>. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the <i>only</i> connected subsets.

</p><p>An important example of a totally disconnected space is the <a href="Cantor_set" title="Cantor set">Cantor set</a>, which is <a href="Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the set of <a href="P-adic_number#p-adic_integers" title="P-adic number"><i>p</i>-adic integers</a>. Another example, playing a key role in <a href="Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, is the field <span class="texhtml"><b>Q</b><sub><i>p</i></sub></span> of <a href="P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>.

</p>

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

<p>A topological space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is <b>totally disconnected</b> if the <a href="Connected_space" title="Connected space">connected components</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> are the one-point sets.<sup id="cite_ref-FOOTNOTERudin1991395_Appendix_A7_1-0" class="reference"><a href="#cite_note-FOOTNOTERudin1991395_Appendix_A7-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMunkres2000152_2-0" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000152-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Analogously, a topological space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is <b>totally path-disconnected</b> if all <a href="Connected_space#Path_connectedness" title="Connected space">path-components</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> are the one-point sets.

</p><p>Another closely related notion is that of a <a href="Totally_separated_space" class="mw-redirect" title="Totally separated space">totally separated space</a>, i.e. a space where <a href="Quasicomponents" class="mw-redirect" title="Quasicomponents">quasicomponents</a> are singletons. That is, a topological space

</mstyle>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is <b>totally separated</b> if for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

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</math></span><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>, the <a href="Intersection" title="Intersection">intersection</a> of all <a href="Clopen" class="mw-redirect" title="Clopen">clopen</a> <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhoods</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is the singleton <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

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

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</math></span><img src="./a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"></span>. Equivalently, for each pair of distinct points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

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

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</math></span><img src="./6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy"></span>, there is a pair of disjoint open neighborhoods <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V}">

</mstyle>

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

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</math></span><img src="./7681409ec5fffdb272f536757c1211fe0151a9b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.604ex; height:2.509ex;" alt="{\displaystyle U,V}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}">

</mstyle>

</mrow>

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

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</math></span><img src="./5ea0abffd33a692ded22accc104515a032851dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=U\sqcup V}">

</mstyle>

</mrow>

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

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</math></span><img src="./da4ddf913641d5b5d70624312fcc67c89a7df753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.231ex; height:2.176ex;" alt="{\displaystyle X=U\sqcup V}" loading="lazy"></span>.

</p><p>Every totally separated space is evidently totally disconnected but the converse is false even for <a href="Metric_space" title="Metric space">metric spaces</a>. For instance, take <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> to be the <a href="Cantor's_teepee" class="mw-redirect" title="Cantor's teepee">Cantor's teepee</a>, which is the <a href="Knaster%E2%80%93Kuratowski_fan" title="Knaster–Kuratowski fan">Knaster–Kuratowski fan</a> with the apex removed. Then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is totally disconnected but its quasicomponents are not singletons. For <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a> the two notions (totally disconnected and totally separated) are equivalent.

</p><p>Confusingly, in the literature<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> totally disconnected spaces are sometimes called <b>hereditarily disconnected</b>,<sup id="cite_ref-FOOTNOTEKuratowski1968151_4-0" class="reference"><a href="#cite_note-FOOTNOTEKuratowski1968151-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> while the terminology <b>totally disconnected</b> is used for totally separated spaces.<sup id="cite_ref-FOOTNOTEKuratowski1968151_4-1" class="reference"><a href="#cite_note-FOOTNOTEKuratowski1968151-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>The following are examples of totally disconnected spaces:

</p>

<ul><li><a href="Discrete_space" title="Discrete space">Discrete spaces</a></li>

<li>The <a href="Rational_number" title="Rational number">rational numbers</a></li>

<li>The <a href="Irrational_number" title="Irrational number">irrational numbers</a></li>

<li>The <i>p</i>-adic numbers; more generally, all <a href="Profinite_group" title="Profinite group">profinite groups</a> are totally disconnected.</li>

<li>The <a href="Cantor_set" title="Cantor set">Cantor set</a> and the <a href="Cantor_space" title="Cantor space">Cantor space</a></li>

<li>The <a href="Baire_space_(set_theory)" title="Baire space (set theory)">Baire space</a></li>

<li>The <a href="Sorgenfrey_line" class="mw-redirect" title="Sorgenfrey line">Sorgenfrey line</a></li>

<li>Every Hausdorff space of <a href="Small_inductive_dimension" class="mw-redirect" title="Small inductive dimension">small inductive dimension</a> 0 is totally disconnected</li>

<li>The <a href="Erd%C5%91s_space" title="Erdős space">Erdős space</a> ℓ<sup><i>2</i></sup><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cap \,\mathbb {Q} ^{\omega }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math></span><img src="./01ff6b51997f512d4c911b0fe649ac7de8cdd6a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.387ex; height:2.676ex;" alt="{\displaystyle \,\cap \,\mathbb {Q} ^{\omega }}" loading="lazy"></span> is a totally disconnected Hausdorff space that does not have small inductive dimension 0.</li>

<li><a href="Extremally_disconnected_space" title="Extremally disconnected space">Extremally disconnected</a> Hausdorff spaces</li>

<li><a href="Stone_space" title="Stone space">Stone spaces</a></li>

<li>The <a href="Knaster%E2%80%93Kuratowski_fan" title="Knaster–Kuratowski fan">Knaster–Kuratowski fan</a> provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.</li></ul>

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

<ul><li><a href="Subspace_(topology)" class="mw-redirect" title="Subspace (topology)">Subspaces</a>, <a href="Product_topology" title="Product topology">products</a>, and <a href="Disjoint_union_(topology)" title="Disjoint union (topology)">coproducts</a> of totally disconnected spaces are totally disconnected.</li>

<li>Totally disconnected spaces are <a href="T1_space" title="T1 space">T<sub>1</sub> spaces</a>, since singletons are closed.</li>

<li>Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every <a href="Compact_space" title="Compact space">compact</a> <a href="Metric_space" title="Metric space">metric space</a> is a continuous image of the <a href="Cantor_set" title="Cantor set">Cantor set</a>.</li>

<li>A <a href="Locally_compact_Hausdorff_space" class="mw-redirect" title="Locally compact Hausdorff space">locally compact Hausdorff space</a> has <a href="Small_inductive_dimension" class="mw-redirect" title="Small inductive dimension">small inductive dimension</a> 0 if and only if it is totally disconnected.</li>

<li>Every totally disconnected compact metric space is homeomorphic to a subset of a <a href="Countable" class="mw-redirect" title="Countable">countable</a> product of <a href="Discrete_space" title="Discrete space">discrete spaces</a>.</li>

<li>It is in general not true that every open set in a totally disconnected space is also closed.</li>

<li>It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected Hausdorff space is <a href="Extremally_disconnected_space" title="Extremally disconnected space">extremally disconnected</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Constructing_a_totally_disconnected_quotient_space_of_any_given_space">Constructing a totally disconnected quotient space of any given space</h2></div>

</mstyle>

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> be an arbitrary topological space. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}">

</mstyle>

</mrow>

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

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</math></span><img src="./bbd1014d850b7c883eb76301dd58c643e3c7e4eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}" loading="lazy"></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathrm {conn} (x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in \mathrm {conn} (x)}</annotation>

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</math></span><img src="./df982891598c4e2a2b154184af36d287fd49a460.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.915ex; height:2.843ex;" alt="{\displaystyle y\in \mathrm {conn} (x)}" loading="lazy"></span> (where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {conn} (x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {conn} (x)}</annotation>

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</math></span><img src="./b85894ee44b912a036d8deddd36dd01625be0c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.919ex; height:2.843ex;" alt="{\displaystyle \mathrm {conn} (x)}" loading="lazy"></span> denotes the largest connected subset containing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>). This is obviously an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> whose equivalence classes are the connected components of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Endow <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/{\sim }}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./80508302af0d1ad574ebaaecbfd8b553e88149d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.951ex; height:2.843ex;" alt="{\displaystyle X/{\sim }}" loading="lazy"></span> with the <a href="Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a>, i.e. the <a href="Finest_topology" class="mw-redirect" title="Finest topology">finest topology</a> making the map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m:x\mapsto \mathrm {conn} (x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m:x\mapsto \mathrm {conn} (x)}</annotation>

</semantics>

</math></span><img src="./a981a4544b41d85e601282b826be9a2253fa44ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.84ex; height:2.843ex;" alt="{\displaystyle m:x\mapsto \mathrm {conn} (x)}" loading="lazy"></span> continuous. With a little bit of effort we can see that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/{\sim }}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./80508302af0d1ad574ebaaecbfd8b553e88149d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.951ex; height:2.843ex;" alt="{\displaystyle X/{\sim }}" loading="lazy"></span> is totally disconnected.

</p><p>In fact this space is not only <i>some</i> totally disconnected quotient but in a certain sense the <i>biggest</i>: The following <a href="Universal_property" title="Universal property">universal property</a> holds: For any totally disconnected space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> and any continuous map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\rightarrow Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b215af1e965d0595a97ad2b21f7d0cbcf6281303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\rightarrow Y}" loading="lazy"></span>, there exists a <i>unique</i> continuous map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}">

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}</annotation>

</semantics>

</math></span><img src="./1f0b9e1c17737650c198e98398f2b91e18cf2229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.429ex; height:3.343ex;" alt="{\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\breve {f}}\circ m}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54f0657445454fc2f01cfbacad264dd5eb17eed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.311ex; height:3.176ex;" alt="{\displaystyle f={\breve {f}}\circ m}" loading="lazy"></span>.

</p>

<ul><li><a href="Extremally_disconnected_space" title="Extremally disconnected space">Extremally disconnected space</a></li>

<li><a href="Totally_disconnected_group" title="Totally disconnected group">Totally disconnected group</a></li></ul>

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

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</style><div class="reflist">

<li id="cite_note-FOOTNOTERudin1991395_Appendix_A7-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERudin1991395_Appendix_A7_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a>, p. 395 Appendix A7.</span>

</li>

<li id="cite_note-FOOTNOTEMunkres2000152-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMunkres2000152_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMunkres2000">Munkres 2000</a>, pp. 152.</span>

</li>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFEngelking1989" class="citation book cs1"><a href="Ryszard_Engelking" title="Ryszard Engelking">Engelking, Ryszard</a> (1989). <i>General Topology</i>. Heldermann Verlag, Sigma Series in Pure Mathematics. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-88538-006-4</bdi>.</cite></span>

</li>

<li id="cite_note-FOOTNOTEKuratowski1968151-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEKuratowski1968151_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKuratowski1968151_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKuratowski1968">Kuratowski 1968</a>, pp. 151.</span>

</li>

</ol></div></div>

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

<ul><li><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). <i>Topology</i> (2nd ed.). <a href="Upper_Saddle_River%2C_NJ" class="mw-redirect" title="Upper Saddle River, NJ">Upper Saddle River, NJ</a>: <a href="Prentice_Hall%2C_Inc" class="mw-redirect" title="Prentice Hall, Inc">Prentice Hall, Inc</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-181629-9</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42683260">42683260</a>.</cite> <span style="font-size:0.95em; font-size:95%; color: var( --color-subtle, #555 )">(<a rel="nofollow" class="external text" href="https://archive.org/details/topology0002edmunk/page/n5/mode/2up">accessible to patrons with print disabilities</a>)</span></li>

<li><cite id="CITEREFRudin1991" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054236-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite></li>

<li><cite id="CITEREFWillard2004" class="citation cs2">Willard, Stephen (2004), <i>General topology</i>, <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43479-7</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2048350">2048350</a></cite> (reprint of the 1970 original, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0264581">0264581</a>)</li>

<li><cite id="CITEREFKuratowski1968" class="citation cs2">Kuratowski, Kazimierz (1968), <i>Topology II: Transl. from French</i> (Revised ed.), New York: Academic Press [u.a.], <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780124292024</bdi></cite></li></ul></div><div><div class="zim-footer">

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