E-5 - Completeness

Abstract

A metric space (X, ρ) is a complete metric space (and ρ is a complete metric) if any decreasing sequence of nonempty closed subsets of X with the diameters converging to zero has nonempty intersections. Compactness implies completeness and a complete metric space (X, ρ) is compact if for each ɛ > 0 it can be covered by finitely many sets with diameters less than ɛ, that is, the metric ρ is totally bounded. A space X is completely metrizable if there is a metric ρ on X generating the topology of X such that the metric space (X, ρ) is complete. Locally compact spaces are Čech-complete and Čech complete spaces are Baire spaces. The Čech -completeness of completely regular spaces is characterized by the existence of a complete sequence of open covers. For metrizable spaces, complete metrizability is equivalent to the existence of a complete sequence of exhaustive covers. A cover-complete (or partition-complete) space is a regular space admitting a complete sequence of exhaustive covers. The spaces that are homeomorphic to closed subsets of products of completely metrizable spaces are called Dieudonné complete.