Editing Complete metric space (section)

==Topologically complete spaces==
Completeness is a property of the ''metric'' and not of the ''topology'', meaning that a complete metric space can be [[homeomorphic]] to a non-complete one.  An example is given by the real numbers, which are complete but homeomorphic to the open interval {{open-open|0,1}}, which is not complete.

In [[topology]] one considers ''[[completely metrizable space]]s'', spaces for which there exists at least one complete metric inducing the given topology.  Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the [[Baire category theorem]] is purely topological, it applies to these spaces as well.

Completely metrizable spaces are often called ''topologically complete''. However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section [[#Alternatives and generalizations|Alternatives and generalizations]]). Indeed, some authors use the term ''topologically complete'' for a wider class of topological spaces, the [[completely uniformizable space]]s.<ref>Kelley, Problem 6.L, p. 208</ref>

A topological space homeomorphic to a [[Separable space|separable]] complete metric space is called a [[Polish space]].