D-12 - Paracompact Spaces

Abstract

As defined by J. Dieudonné in 1944, a topological space X is paracompact if it is Hausdorff and if every open cover ▪ of X has a locally finite, open refinement ν. There are numerous characterizations of paracompact spaces; many of them are in terms of different kinds of refinements of open covers. All compact Hausdorff spaces are paracompact, and more generally, all regular Lindelöf spaces are also paracompact. In the opposite direction, all paracompact spaces are normal. Stone's theorem states that every metrizable space is paracompact. The Nagata–Smirnov Metrization theorem states that a regular space is metrizable if and only if (iff) it has a σ -locally finite base. A regular space X is paracompact iff every open cover of X has a σ -locally finite, open refinement. The behavior of paracompact spaces under various operations is generally similar to that of normal spaces. The behavior of paracompact spaces under various operations is generally similar to that of normal spaces, with a significant exception: Paracompact spaces are preserved by products with compact Hausdorff spaces, whereas normal spaces are not.