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Concept
Paracompact space
Summary
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.
The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact. Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact space and a compact space is always paracompact.
Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space.
A cover of a set is a collection of subsets of whose union contains . In symbols, if is an indexed family of subsets of , then is a cover of if
A cover of a topological space is open if all its members are open sets. A refinement of a cover of a space is a new cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols, the cover is a refinement of the cover if and only if, for every in , there exists some in such that .
An open cover of a space is locally finite if every point of the space has a neighborhood that intersects only finitely many sets in the cover.
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