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| In [[topology]] and related areas of [[mathematics]], a '''metrizable space''' is a [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[metric space]]. That is, a topological space <math>(X,\tau)</math> is said to be metrizable if there is a metric
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| :<math>d\colon X \times X \to [0,\infty)</math>
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| such that the topology induced by ''d'' is <math>\tau</math>. '''Metrization theorems''' are [[theorem]]s that give [[sufficient condition]]s for a topological space to be metrizable.
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| ==Properties==
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| Metrizable spaces inherit all topological properties from metric spaces. For example, they are [[Hausdorff space|Hausdorff]] [[paracompact]] spaces (and hence [[normal space|normal]] and [[Tychonoff space|Tychonoff]]) and [[first-countable space|first-countable]]. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable [[uniform space]], for example, may have a different set of [[Contraction_mapping|contraction maps]] than a metric space to which it is homeomorphic.
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| ==Metrization theorems==
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| One of the first widely-recognized metrization theorems was '''Urysohn's metrization theorem'''. This states that every Hausdorff [[second-countable]] [[regular space]] is metrizable. So, for example, every second-countable [[manifold]] is metrizable. (Historical note: The form of the theorem shown here was in fact proved by [[Andrey Nikolayevich Tychonoff|Tychonoff]] in 1926. What [[Pavel Samuilovich Urysohn|Urysohn]] had shown, in a paper published posthumously in 1925, was that every second-countable ''[[normal space|normal]]'' 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.<ref>http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf</ref> The [[Nagata–Smirnov metrization theorem]], described below, provides a more specific theorem where the converse does hold.
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| Several other metrization theorems follow as simple corollaries to Urysohn's Theorem. For example, a [[Compact space|compact]] Hausdorff space is metrizable if and only if it is second-countable.
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| Urysohn's Theorem can be restated as: A topological space is [[separable space|separable]] 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 if and only if 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 [[locally finite collection]]s of open sets. For a closely related theorem see the [[Bing metrization theorem]].
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| Separable metrizable spaces can also be characterized as those spaces which are [[homeomorphic]] to a subspace of the [[Hilbert cube]] <math>\lbrack 0,1\rbrack ^\mathbb{N}</math>, i.e. the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the [[product topology]].
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| A space is said to be '''locally metrizable''' if every point has a metrizable [[neighbourhood (mathematics)|neighbourhood]]. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and [[paracompact]]. In particular, a manifold is metrizable if and only if it is paracompact.
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| ==Examples==
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| The group of unitary operators <math> \mathbb{U}(\mathcal{H})</math> on a separable Hilbert space <math> \mathcal{H}</math> endowed
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| with the strong operator topology is metrizable (see Proposition II.1 in <ref>Neeb, Karl-Hermann,
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| On a theorem of S. Banach.
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| J. Lie Theory 7 (1997), no. 2, 293–300. </ref>).
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| == Examples of non-metrizable spaces==
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| Non-normal spaces cannot be metrizable; important examples include
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| * the [[Zariski topology]] on an [[algebraic variety]] or on the [[spectrum of a ring]], used in [[algebraic geometry]],
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| * the [[topological vector space]] of all [[function (mathematics)|function]]s from the [[real line]] '''R''' to itself, with the [[topology of pointwise convergence]].
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| The real line with the [[lower limit topology]] 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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| The [[long line (topology)|long line]] is locally metrizable but not metrizable; in a sense it is "too long".
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| == See also ==
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| * [[Uniformizability]], the property of a topological space of being homeomorphic to a [[uniform space]], or equivalently the topology being defined by a family of [[pseudometric]]s
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| * [[Moore space (topology)]]
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| * [[Ion Barbu#Apollonian metric|Apollonian metric]]
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| ==References==
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| {{reflist}}
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| {{PlanetMath attribution|id=1538|title=Metrizable}}
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| [[Category:General topology]]
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| [[Category:Theorems in topology]]
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