# 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).

## Contents

### 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.