Editing Metrizable space (section)

==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.