285:
However, no confusion can arise when talking about an abelian topological group or a topological vector space being completely metrizable: it can be proven that every abelian topological group (and thus also every topological vector space) that is completely metrizable as a topological space (i. e.,
129:(the latter would yield the definition of complete metric space). Once we make the choice of the metric on a completely metrizable space (out of all the complete metrics compatible with the topology), we get a complete metric space. In other words, the
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of that of topological spaces, while the category of complete metric spaces is not (instead, it is a subcategory of the category of metric spaces). Complete metrizability is a topological property while completeness is a property of the metric.
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if and only if each factor is completely metrizable. Hence, a product of nonempty metrizable spaces is completely metrizable if and only if at most countably many factors have more than one point and each factor is completely
261:. In general, there are many such completely metrizable spaces, since completions of a topological space with respect to different metrics compatible with its topology can give topologically different completions.
274:, the natural meaning of the words ācompletely metrizableā would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology. For
293:(induced by its topology and addition operation) is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.
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Because a product of nonempty metrizable spaces is metrizable if and only if at most countably many factors have more than one point (Willard, Chapter 22).
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This implies e. g. that every completely metrizable topological vector space is complete. Indeed, a topological vector space is called complete iff its
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For every metrizable space there exists a completely metrizable space containing it as a dense subspace, since every metric space has a
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admits a complete metric that induces its topology) also admits an invariant complete metric that induces its topology.
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A countable product of nonempty metrizable spaces is completely metrizable in the
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in the definition of completely metrizable space, which is not the same as
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When talking about spaces with more structure than just topology, like
444:"Invariant metrics in groups (solution of a problem of Banach)"
348:
e. g. Steen and
Seebach, I Ā§5: Complete Metric Spaces
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266:Completely metrizable abelian topological groups
233:is completely metrizable if and only if it is
160:, but it is completely metrizable since it is
86:is employed by some authors as a synonym for
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229:A subspace of a completely metrizable space
189:is metrizable but not completely metrizable.
183:with the subspace topology inherited from
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203:is completely metrizable if and only if
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37:metrically topologically complete space
51:) for which there exists at least one
540:. Addison-Wesley Publishing Company.
133:of completely metrizable spaces is a
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517:. Holt, Rinehart and Winston, Inc.
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464:10.1090/s0002-9939-1952-0047250-4
123:there exists at least one metric
92:completely uniformizable spaces
308:Completely uniformizable space
1:
84:topologically complete space
514:Counterexamples in Topology
357:Kelley, Problem 6.L, p. 208
339:Kelley, Problem 6.K, p. 207
220:StoneāÄech compactification
115:completely metrizable space
108:completely metrizable space
88:completely metrizable space
33:completely metrizable space
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113:The distinction between a
532:Willard, Stephen (1970).
280:topological vector spaces
330:Willard, Definition 24.2
278:topological groups and
127:there is given a metric
509:Seebach, J. Arthur Jr.
396:Willard, Theorem 24.13
451:Proc. Amer. Math. Soc
387:Willard, Exercise 25A
303:Complete metric space
119:complete metric space
104:complete metric space
78:induces the topology
72:complete metric space
18:Completely metrizable
442:Klee, V. L. (1952).
199:A topological space
96:Äech-complete spaces
432:Willard, Chapter 24
414:Willard, Chapter 24
405:Willard, Chapter 24
378:Willard, Chapter 24
102:Difference between
505:Steen, Lynn Arthur
272:topological groups
121:lies in the words
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524:978-0-03-079485-8
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369:Section 24.
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135:subcategory
82:. The term
62:such that (
29:mathematics
475:References
291:uniformity
259:completion
209:metrizable
194:Properties
173:The space
147:The space
560:Category
511:(1970).
485:(1975).
297:See also
149:(0,1) ā
142:Examples
131:category
276:abelian
218:in its
70:) is a
39:) is a
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211:and a
117:and a
53:metric
447:(PDF)
319:Notes
542:ISBN
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106:and
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