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completely metrizable space
In mathematics, a completely metrizable space〔Willard, Definition 24.2〕 (metrically topologically complete space〔Kelley, Problem 6.K, p. 207〕) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' induces the topology ''T''. The term topologically complete space is employed by some authors as a synonym for ''completely metrizable space'',〔e. g. Steen and Seebach, I §5: Complete Metric Spaces〕 but sometimes also used for other classes of topological spaces, like completely uniformizable spaces〔Kelley, Problem 6.L, p. 208〕 or Čech-complete spaces.
==Difference between ''complete metric space'' and ''completely metrizable space''==
The difference between ''completely metrizable space'' and ''complete metric space'' is in the words ''there exists at least one metric'' in the definition of completely metrizable space, which is not the same as ''there is given a metric'' (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 category of completely metrizable spaces is a subcategory 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.〔 Section 24.〕
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