View source for Metrizable space

{{short description|Topological space that is homeomorphic to a metric space}}
In [[Topology|topology]] and related areas of [[Mathematics|mathematics]], a '''metrizable space''' is a [[Topological space|topological space]] that is [[Homeomorphism|homeomorphic]] to a [[Metric space|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|title=Topology|year=1999|publisher=Pearson|page=119|edition=second}}</ref> '''Metrization theorems''' are [[Theorem|theorem]]s that give sufficient conditions 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 completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable [[Uniform space|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|regular space]] is metrizable. So, for example, every second-countable [[Manifold|manifold]] is metrizable. (Historical note: The form of the theorem shown here was in fact proved by Tikhonov in 1926. What 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=Archived copy|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}}</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|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|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|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|algebraic variety]] or on the [[Spectrum of a ring|spectrum of a ring]], used in [[Algebraic geometry|algebraic geometry]],
* the [[Topological vector space|topological vector space]] of all [[Function (mathematics)|function]]s from the [[Real line|real line]] <math>\R</math> to itself, with the topology of pointwise convergence.

The real line with the [[Lower limit topology|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|non-Hausdorff manifold]] (and thus cannot be metrizable). Like all manifolds, it is locally homeomorphic to [[Euclidean space]] and thus 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|locally regular space]] but not a [[Semiregular space|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|Bing metrization theorem}}
* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Moore space (topology)}}
* {{annotated link|Nagata–Smirnov metrization theorem}}

==References==

{{reflist}}


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

{{Sourceattribution|Metrizable space}}