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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
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
:<math>d\colon X \times X \to [0,\infty)</math>


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.


==Metrization theorems==
'''MathML'''
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.
:<math forcemathmode="mathml">E=mc^2</math>


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.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


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]].
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


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]].
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


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.
==Demos==


==Examples==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
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]] '''R''' to itself, with the [[topology of pointwise convergence]].


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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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.
==Test pages ==


The [[long line (topology)|long line]] is locally metrizable but not metrizable; in a sense it is "too long".
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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== See also ==
*[[Inputtypes|Inputtypes (private Wikis only)]]
* [[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
*[[Url2Image|Url2Image (private Wikis only)]]
* [[Moore space (topology)]]
==Bug reporting==
* [[Ion Barbu#Apollonian metric|Apollonian metric]]
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
*  [[Nagata–Smirnov metrization theorem]]
* [[Bing metrization theorem]]
 
==References==
{{reflist}}
 
{{PlanetMath attribution|id=1538|title=Metrizable}}
 
[[Category:General topology]]
[[Category:Theorems in topology]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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