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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
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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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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. The
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that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the
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Several other metrization theorems follow as simple corollaries to
Urysohn's theorem. For example, a
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443:. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and
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Metrizable spaces inherit all topological properties from metric spaces. For example, they are
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is metrizable. (Historical note: The form of the theorem shown here was in fact proved by
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Neeb, Karl-Hermann, On a theorem of S. Banach. J. Lie Theory 7 (1997), no. 2, 293–300.
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had shown, in a paper published posthumously in 1925, was that every second-countable
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and metrizable if and only if it is regular, Hausdorff and second-countable. The
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Separable metrizable spaces can also be characterized as those spaces which are
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703: – Topological space whose topology is generated by a uniform structure
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is locally metrizable but not metrizable; in a sense it is "too long".
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Hausdorff space is metrizable if and only if it is second-countable.
781:"Math 395 - Honors Analysis I: 10. Some counterexamples in topology"
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Non-normal spaces cannot be metrizable; important examples include
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101:
707:, the property of a topological space of being homeomorphic to a
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Urysohn's
Theorem can be restated as: A topological space is
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One of the first widely recognized metrization theorems was
617:(and thus cannot be metrizable). Like all manifolds, it is
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Topological space that is homeomorphic to a metric space
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Pages displaying short descriptions of redirect targets
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is metrizable. So, for example, every second-countable
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This article incorporates material from
Metrizable on
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Pages displaying wikidata descriptions as a fallback
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of open sets. For a closely related theorem see the
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302:than a metric space to which it is homeomorphic.
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824:Creative Commons Attribution/Share-Alike License
57:but its sources remain unclear because it lacks
483:{\displaystyle \mathbb {U} ({\mathcal {H}})}
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688: – developable regular Hausdorff space
298:, for example, may have a different set of
518:is metrizable (see Proposition II.1 in ).
263:for a topological space to be metrizable.
202:{\displaystyle d:X\times X\to [0,\infty )}
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88:Learn how and when to remove this message
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670: – Romanian mathematician and poet
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156:is said to be metrizable if there is a
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599:Locally metrizable but not metrizable
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680:Metrizable topological vector space
318:. This states that every Hausdorff
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209:such that the topology induced by
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584:topology of pointwise convergence
522:Examples of non-metrizable spaces
855:Properties of topological spaces
439:if every point has a metrizable
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779:Mitya Boyarchenko (Fall 2010).
455:The group of unitary operators
124:. That is, a topological space
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368:locally finite collections
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674:Bing metrization theorem
551:topological vector space
516:strong operator topology
372:Bing metrization theorem
43:This article includes a
18:Locally metrizable space
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734:"Metrization Theorems"
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45:list of references
732:Simon, Jonathan.
668:Apollonian metric
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379:homeomorphic
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64:Please help
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445:paracompact
276:paracompact
106:mathematics
70:introducing
834:Categories
820:PlanetMath
797:2012-08-08
719:References
647:but not a
542:, used in
538:or on the
267:Properties
259:that give
845:Manifolds
656:long line
635:Hausdorff
633:(but not
625:and thus
559:real line
557:from the
555:functions
359:separable
284:Tychonoff
273:Hausdorff
237:τ
194:∞
182:→
176:×
141:τ
762:Topology
760:(1999).
662:See also
451:Examples
331:Tikhonov
327:manifold
257:theorems
116:that is
102:topology
766:Pearson
742:16 June
553:of all
352:compact
335:Urysohn
66:improve
534:on an
340:normal
286:) and
280:normal
158:metric
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613:is a
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