Topologically Complete Spaces
Note that 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 (0, 1), which is not complete. Another example is given by the irrational numbers, which are not complete as a subspace of the real numbers but are homeomorphic to NN (see the sequence example in Examples above).
In topology one considers topologically complete (or completely metrizable) spaces, 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.
A topological space homeomorphic to a separable complete metric space is called a Polish space.
Read more about this topic: Complete Metric Space
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—Jane Austen (1775–1817)
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