Paracompact space

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In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by Dieudonné (1944).


Definitions of relevant terms

  • A cover of a set X is a collection of subsets of X whose union is X. In symbols, if U = {Uα : α in A} is an indexed family of subsets of X, then U is a cover if and only if
  • A cover of a topological space X is open if all its members are open sets. In symbols, a cover U is an open cover if U is a subset of T, where T is the topology on X.
  • A refinement of a cover of a space X 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 V = {Vβ : β in B} is a refinement of the cover U = {Uα : α in A} if and only if, for any Vβ in V, there exists some Uα in U such that Vβ is contained in Uα.
  • An open cover of a space X is locally finite if every point of the space has a neighborhood which intersects only finitely many sets in the cover. In symbols, U = {Uα : α in A} is locally finite if and only if, for any x in X, there exists some neighbourhood V(x) of x such that the set
  • A topological space X is called paracompact if any open cover of X admits an open refinement that is locally finite.

Note the similarity between the definitions of compact and paracompact: for paracompact, we replace "subcover" by "open refinement" and "finite" by "locally finite". Both of these changes are significant: if we take the above definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases.

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