View source for Metrizable space

{{short description|Topological space that is homeomorphic to a metric space}}
{{inline citations|date=September 2024}}
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 (mathematics)|metric]] <math>d : X \times X \to [0, \infty)</math> such that the topology induced by <math>d</math> is <math>\tau.</math><ref>{{cite web|last=Simon|first=Jonathan|title=Metrization Theorems|url=http://homepage.math.uiowa.edu/~jsimon/COURSES/M132Fall07/MetrizationTheorem_v5.pdf|access-date=16 June 2016}}</ref><ref>{{cite book|last=Munkres|first=James|author-link=James Munkres|title=Topology|year=1999|publisher=[[Pearson PLC|Pearson]]|page=119|edition=second}}</ref> '''Metrization theorems''' are [[theorem]]s that give [[sufficient condition]]s for a topological space to be metrizable.

==Properties==

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 [[Complete metric space|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.

==Metrization theorems==

One of the first widely recognized metrization theorems was '''{{visible anchor|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|Tikhonov]] 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>{{Cite web|url=http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf|title=Math 395 - Honors Analysis I: 10. Some counterexamples in topology |date=Fall 2010|access-date=2012-08-08|archive-url=https://web.archive.org/web/20110925003841/http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf|archive-date=2011-09-25|url-status=dead |author=Mitya Boyarchenko}}</ref> The [[Nagata–Smirnov metrization theorem]], described below, provides a more specific theorem where the converse does hold.

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.

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]].

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 ^\N,</math> that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the [[product topology]].

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.

==Examples==

The group of unitary operators <math>\mathbb{U}(\mathcal{H})</math> on a separable Hilbert space <math>\mathcal{H}</math> endowed
with the [[strong operator topology]] is metrizable (see Proposition II.1 in <ref>Neeb, Karl-Hermann, On a theorem of S. Banach. J. Lie Theory 7 (1997), no. 2, 293–300.</ref>).

'''Examples of non-metrizable spaces'''

Non-normal spaces cannot be metrizable; important examples include
* the [[Zariski topology]] on an [[algebraic variety]] or on the [[spectrum of a ring]], used in [[algebraic geometry]],
* the [[topological vector space]] of all [[Function (mathematics)|function]]s from the [[real line]] <math>\R</math> to itself, with the [[topology of pointwise convergence]].

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.

===Locally metrizable but not metrizable===

The [[Line with two origins]], also called the ''{{dfn|bug-eyed line}}'' is a [[non-Hausdorff manifold]] (and thus cannot be metrizable). Like all manifolds, it is [[locally homeomorphic]] to [[Euclidean space]] and thus [[locally metrizable space|locally metrizable]] (but not metrizable) and [[Locally Hausdorff space|locally Hausdorff]] (but not [[Hausdorff space|Hausdorff]]). It is also a [[T1 space|T<sub>1</sub>]] [[locally regular space]] but not a [[semiregular space]].

The [[Long line (topology)|long line]] is locally metrizable but not metrizable; in a sense it is "too long".

==See also==

* {{annotated link|Ion Barbu#Apollonian metric|Apollonian metric}}
* {{annotated link|Bing metrization theorem}}
* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Moore space (topology)}}
* {{annotated link|Nagata–Smirnov metrization theorem}}
* {{annotated link|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 space|pseudometrics]]

==References==

{{reflist}}

{{Metric spaces}}
{{PlanetMath attribution|id=1538|title=Metrizable}}

[[Category:General topology]]
[[Category:Manifolds]]
[[Category:Metric spaces]]
[[Category:Properties of topological spaces]]
[[Category:Theorems in topology]]
[[Category:Topological spaces]]