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<title>Metrizable space</title>

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

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<p>In <a href="Topology" title="Topology">topology</a> and related areas of <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>metrizable space</b> is a <a href="Topological_space" title="Topological space">topological space</a> that is <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a> to a <a href="Metric_space" title="Metric space">metric space</a>. That is, 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,\tau )}">

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</math></span><img src="./dede4b8004c6222d11e9b1a9802dc0496ad7e1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}" loading="lazy"></span> is said to be metrizable if there is a <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d:X\times X\to [0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle d:X\times X\to [0,\infty )}</annotation>

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</math></span><img src="./ee12b40a6905ef3bdf7a2e087fd9905aa882d95b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.639ex; height:2.843ex;" alt="{\displaystyle d:X\times X\to [0,\infty )}" loading="lazy"></span> such that the topology induced by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math></span><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"></span><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><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> <i>Metrization theorems</i> are <a href="Theorem" title="Theorem">theorems</a> that give <a href="Sufficient_condition" class="mw-redirect" title="Sufficient condition">sufficient conditions</a> for a topological space to be metrizable.

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

<p>Metrizable spaces inherit all topological properties from metric spaces. For example, they are <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="Paracompact" class="mw-redirect" title="Paracompact">paracompact</a> spaces (and hence <a href="Normal_space" title="Normal space">normal</a> and <a href="Tychonoff_space" title="Tychonoff space">Tychonoff</a>) and <a href="First-countable_space" title="First-countable space">first-countable</a>. However, some properties of the metric, such as <a href="Complete_metric_space" title="Complete metric space">completeness</a>, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable <a href="Uniform_space" title="Uniform space">uniform space</a>, for example, may have a different set of <a href="Contraction_mapping" title="Contraction mapping">contraction maps</a> than a metric space to which it is homeomorphic.

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<div class="mw-heading mw-heading2"><h2 id="Metrization_theorems">Metrization theorems</h2></div>

<p>One of the first widely recognized metrization theorems was <i><style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style><span class="vanchor"><span class="vanchor-text">Urysohn's metrization theorem</span></span></i>. This states that every Hausdorff <a href="Second-countable" class="mw-redirect" title="Second-countable">second-countable</a> <a href="Regular_space" title="Regular space">regular space</a> is metrizable. So, for example, every second-countable <a href="Manifold" title="Manifold">manifold</a> is metrizable. (Historical note: The form of the theorem shown here was in fact proved by <a href="Andrey_Nikolayevich_Tychonoff" class="mw-redirect" title="Andrey Nikolayevich Tychonoff">Tikhonov</a> in 1926. What <a href="Pavel_Samuilovich_Urysohn" class="mw-redirect" title="Pavel Samuilovich Urysohn">Urysohn</a> had shown, in a paper published posthumously in 1925, was that every second-countable <i><a href="Normal_space" title="Normal space">normal</a></i> Hausdorff space is metrizable.) The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.<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> The <a href="Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov metrization theorem</a>, described below, provides a more specific theorem where the converse does hold.

</p><p>Several other metrization theorems follow as simple corollaries to Urysohn's theorem. For example, a <a href="Compact_space" title="Compact space">compact</a> Hausdorff space is metrizable if and only if it is second-countable.

</p><p>Urysohn's Theorem can be restated as: A topological space is <a href="Separable_space" title="Separable space">separable</a> and metrizable if and only if it is regular, Hausdorff and second-countable. The Nagata–Smirnov metrization theorem extends this to the non-separable case. It states that a topological space is metrizable <a href="If_and_only_if" title="If and only if">if and only if</a> it is regular, Hausdorff and has a σ-locally finite base. A σ-locally finite base is a base which is a union of countably many <a href="Locally_finite_collection" title="Locally finite collection">locally finite collections</a> of open sets. For a closely related theorem see the <a href="Bing_metrization_theorem" title="Bing metrization theorem">Bing metrization theorem</a>.

</p><p>Separable metrizable spaces can also be characterized as those spaces which are <a href="Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to a subspace of the <a href="Hilbert_cube" title="Hilbert cube">Hilbert cube</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrack 0,1\rbrack ^{\mathbb {N} },}">

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</math></span><img src="./f26c2129cca2e9ab8baee625f022ec82dfc7f678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:3.176ex;" alt="{\displaystyle \lbrack 0,1\rbrack ^{\mathbb {N} },}" loading="lazy"></span> that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the <a href="Product_topology" title="Product topology">product topology</a>.

</p><p>A space is said to be <i>locally metrizable</i> if every point has a metrizable <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a>. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and <a href="Paracompact" class="mw-redirect" title="Paracompact">paracompact</a>. In particular, a manifold is metrizable if and only if it is paracompact.

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

<p>The group of unitary operators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} ({\mathcal {H}})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} ({\mathcal {H}})}</annotation>

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</math></span><img src="./3bd95fd7c79db6175d220d69287f8ed3a0277feb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.451ex; height:2.843ex;" alt="{\displaystyle \mathbb {U} ({\mathcal {H}})}" loading="lazy"></span> on a separable Hilbert 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 {\mathcal {H}}}">

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

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</math></span><img src="./19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span> endowed

with the <a href="Strong_operator_topology" title="Strong operator topology">strong operator topology</a> is metrizable (see Proposition II.1 in <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>).

</p><p>Non-normal spaces cannot be metrizable; important examples include

</p>

<ul><li>the <a href="Zariski_topology" title="Zariski topology">Zariski topology</a> on an <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a> or on the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a>, used in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>,</li>

<li>the <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> of all <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> from the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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

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</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> to itself, with the <a href="Topology_of_pointwise_convergence" class="mw-redirect" title="Topology of pointwise convergence">topology of pointwise convergence</a>.</li></ul>

<p>The real line with the <a href="Lower_limit_topology" title="Lower limit topology">lower limit topology</a> is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.

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<div class="mw-heading mw-heading3"><h3 id="Locally_metrizable_but_not_metrizable">Locally metrizable but not metrizable</h3></div>

<p>The <a href="Line_with_two_origins" class="mw-redirect" title="Line with two origins">Line with two origins</a>, also called the <i><dfn>bug-eyed line</dfn></i> is a <a href="Non-Hausdorff_manifold" title="Non-Hausdorff manifold">non-Hausdorff manifold</a> (and thus cannot be metrizable). Like all manifolds, it is <a href="Locally_homeomorphic" class="mw-redirect" title="Locally homeomorphic">locally homeomorphic</a> to <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> and thus <a href="Locally_metrizable_space" class="mw-redirect" title="Locally metrizable space">locally metrizable</a> (but not metrizable) and <a href="Locally_Hausdorff_space" title="Locally Hausdorff space">locally Hausdorff</a> (but not <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>). It is also a <a href="T1_space" title="T1 space">T<sub>1</sub></a> <a href="Locally_regular_space" class="mw-redirect" title="Locally regular space">locally regular space</a> but not a <a href="Semiregular_space" title="Semiregular space">semiregular space</a>.

</p><p>The <a href="Long_line_(topology)" title="Long line (topology)">long line</a> is locally metrizable but not metrizable; in a sense, it is "too long".

</p>

<ul><li><a href="Ion_Barbu#Apollonian_metric" title="Ion Barbu">Apollonian metric</a> – Romanian mathematician and poet (1895 - 1961)</li>

<li><a href="Bing_metrization_theorem" title="Bing metrization theorem">Bing metrization theorem</a> – Characterizes when a topological space is metrizable</li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable topological vector space</a> – Topological vector space whose topology can be defined by a metric</li>

<li><a href="Moore_space_(topology)" title="Moore space (topology)">Moore space (topology)</a></li>

<li><a href="Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov metrization theorem</a> – Characterizes when a topological space is metrizable</li>

<li><a href="Uniformizability" class="mw-redirect" title="Uniformizability">Uniformizability</a> – Topological space whose topology is generated by a uniform structure<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span>, the property of a topological space of being homeomorphic to a <a href="Uniform_space" title="Uniform space">uniform space</a>, or equivalently the topology being defined by a family of <a href="Pseudometric_space" title="Pseudometric space">pseudometrics</a></li></ul>

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

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</style><cite id="CITEREFSimon" class="citation web cs1">Simon, Jonathan. <a rel="nofollow" class="external text" href="http://homepage.math.uiowa.edu/~jsimon/COURSES/M132Fall07/MetrizationTheorem_v5.pdf">"Metrization Theorems"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">16 June</span> 2016</span>.</cite></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"><cite id="CITEREFMunkres1999" class="citation book cs1"><a href="James_Munkres" title="James Munkres">Munkres, James</a> (1999). <i>Topology</i> (second ed.). <a href="Pearson_PLC" class="mw-redirect" title="Pearson PLC">Pearson</a>. p. 119.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFMitya_Boyarchenko2010" class="citation web cs1">Mitya Boyarchenko (Fall 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110925003841/http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf">"Math 395 - Honors Analysis I: 10. Some counterexamples in topology"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-09-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-08-08</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Neeb, Karl-Hermann, On a theorem of S. Banach. J. Lie Theory 7 (1997), no. 2, 293–300.</span>

</li>

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</style><div id="Metric_spaces_(Category)86" style="font-size:114%;margin:0 4em"><a href="Metric_space" title="Metric space">Metric spaces</a> (Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic 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="Metric_space" title="Metric space">Metric space</a></li>

<li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Complete_metric_space" title="Complete metric space">Completeness</a></li>

<li><a href="Equivalence_of_metrics" title="Equivalence of metrics">Equivalent metrics</a></li>

<li><a href="Triangle_inequality" title="Triangle inequality">Triangle inequality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main 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="Baire_category_theorem" title="Baire category theorem">Baire category theorem</a></li>

<li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point</a></li>

<li><a href="Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a></li>

<li><a href="Lebesgue's_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a></li>

<li><a href="Metrization_theorem" class="mw-redirect" title="Metrization theorem">Metrization theorems</a>:

<li><a href="Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov</a></li>

<li><a href="Urysohn's_metrization_theorem" class="mw-redirect" title="Urysohn's metrization theorem">Urysohn's</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</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="Contraction_mapping" title="Contraction mapping">Contraction</a>

<ul><li><a href="Metric_map" title="Metric map">Metric map</a></li></ul></li>

<li><a href="Dilation_(metric_space)" title="Dilation (metric space)">Dilation</a></li>

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

<li>(<a href="Quasi-isometry" title="Quasi-isometry">Quasi-</a>) <a href="Isometry" title="Isometry">Isometry</a></li>

<li><a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a></li>

<li><a href="Metric_derivative" title="Metric derivative">Metric derivative</a></li>

<li><a href="Metric_outer_measure" title="Metric outer measure">Metric outer measure</a></li>

<li><a href="Metric_projection" title="Metric projection">Metric projection</a></li>

<li><a href="Motion_(geometry)" title="Motion (geometry)">Motion</a></li>

<li><a href="Quasisymmetric_map" title="Quasisymmetric map">Quasisymmetric</a></li>

<li><a href="Stretch_factor" title="Stretch factor">Stretch factor</a></li>

<li><a href="Uniform_continuity" title="Uniform continuity">Uniform continuity</a>

<ul><li><a href="Uniform_isomorphism" title="Uniform isomorphism">Isomorphism</a></li></ul></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform convergence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br>metric spaces</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="Complete_metric_space" title="Complete metric space">Complete</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex</a></li>

<li><a href="Doubling_space" title="Doubling space">Doubling</a></li>

<li><a href="Hyperbolic_metric_space" title="Hyperbolic metric space">Hyperbolic</a></li>

<li><a href="Injective_metric_space" title="Injective metric space">Injective</a></li>

<li><a href="Length_metric_space" class="mw-redirect" title="Length metric space">Length metric space</a></li>

<li><a href="Metric_space_aimed_at_its_subspace" title="Metric space aimed at its subspace">Metric space aimed at its subspace</a></li>

<li><a href="Polish_space" title="Polish space">Polish</a></li>

<li><a href="Totally_bounded_space" title="Totally bounded space">Totally bounded</a></li>

<li><a href="Tree-graded_space" title="Tree-graded space">Tree-graded</a></li>

<li><a href="Ultrametric_space" title="Ultrametric space">Ultrametric space</a></li>

<li><a href="Uniformly_disconnected_space" title="Uniformly disconnected space">Uniformly disconnected</a></li>

<li><a href="Urysohn_universal_space" title="Urysohn universal space">Urysohn universal</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</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="Ball_(mathematics)" title="Ball (mathematics)">Balls</a></li>

<li><a href="Borel_set" title="Borel set">Borel</a></li>

<li><a href="Bounded_set" title="Bounded set">Bounded</a></li>

<li><a href="Delone_set" title="Delone set">Delone</a></li>

<li><a href="Diameter_of_a_set" title="Diameter of a set">Diameter</a></li>

<li><a href="Distance_set" title="Distance set">Distance set</a></li>

<li><a href="Gromov_product" title="Gromov product">Gromov product</a></li>

<li><a href="Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff convergence</a></li>

<li><a href="Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a></li>

<li><a href="Kuratowski_convergence" title="Kuratowski convergence">Kuratowski convergence</a></li>

<li><a href="Meyer_set" title="Meyer set">Meyer</a></li>

<li><a href="Packing_dimension" title="Packing dimension">Packing dimension</a></li>

<li><a href="Porous_set" title="Porous set">Porous</a></li>

<li><a href="Positively_separated_sets" title="Positively separated sets">Positively separated sets</a></li>

<li><a href="Tight_span" title="Tight span">Tight span</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Manifold" title="Manifold">Manifolds</a></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="Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a><br>and <a href="Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></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="Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li>

<li><a href="Inner_product_space" title="Inner product space">Inner product space</a></li>

<li><a href="L%C3%A9vy_metric" title="Lévy metric">Lévy metric</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable topological vector space</a></li>

<li><a href="Normed_space" class="mw-redirect" title="Normed space">Normed space</a></li>

<li><a href="Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li>

<li><a href="Wasserstein_metric" title="Wasserstein metric">Wasserstein metric</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="General_topology" title="General topology">General topology</a></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="Discrete_space" title="Discrete space">Discrete space</a></li>

<li><a href="Intrinsic_metric" title="Intrinsic metric">Intrinsic metric</a></li>

<li><a href="Laakso_space" title="Laakso space">Laakso space</a></li>

<li><a href="Product_metric" title="Product metric">Product metric</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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="Category_of_metric_spaces" title="Category of metric spaces">Category of metric spaces</a></li>

<li><a href="Cantor_space" title="Cantor space">Cantor space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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="Approach_space" title="Approach space">Approach space</a></li>

<li><a href="Cauchy_space" title="Cauchy space">Cauchy space</a></li>

<li><a href="Coarse_structure" title="Coarse structure">Coarse structure</a></li>

<li><a href="Cosmic_space" title="Cosmic space">Cosmic space</a></li>

<li><a href="Diversity_(mathematics)" title="Diversity (mathematics)">Diversity</a></li>

<li><a href="Generalised_metric" title="Generalised metric">Generalised metric</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a></li>

<li><a href="Probabilistic_metric_space" title="Probabilistic metric space">Probabilistic metric space</a></li>

<li><a href="Proximity_space" title="Proximity space">Proximity space</a></li>

<li><a href="Pseudometric_space" title="Pseudometric space">Pseudometric space</a></li>

<li><a href="Uniform_space" title="Uniform space">Uniform space</a></li></ul>

</div></td></tr></tbody></table></div>

<p><i>This article incorporates material from Metrizable on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>

</p></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-10" href="https://en.wikipedia.org/wiki/?title=Metrizable_space&oldid=1284953160">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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