Editing Paracompact space (section)

== Definition ==

A ''[[cover (set theory)|cover]]'' of a [[Set (mathematics)|set]] <math>X</math> is a collection of [[subset]]s of <math>X</math> whose [[union (set theory)|union]] contains <math>X</math>. In symbols, if <math>U = \{U_\alpha : \alpha \in A\}</math> is an [[indexed family]] of subsets of <math>X</math>, then <math>U</math> is a cover of <math>X</math> if

: <math>X \subseteq \bigcup_{\alpha \in A}U_{\alpha}.</math>

A cover of a topological space <math>X</math> is ''[[open cover|open]]'' if all its members are [[open set]]s. A ''refinement'' of a cover of a space <math>X</math> 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 <math>V = \{V_\beta : \beta \in B\}</math> is a refinement of the cover <math>U = \{U_\alpha : \alpha \in A\}</math> if and only if, [[universal quantification|for every]] <math>V_\beta</math> in <math>V</math>, [[existential quantification|there exists some]] <math>U_\alpha</math> in <math>U</math> such that <math>V_\beta \subseteq U_\alpha</math>.

An open cover of a space <math>X</math> is ''locally finite'' if every point of the space has a [[neighborhood (topology)|neighborhood]] that intersects only [[finite set|finite]]ly many sets in the cover. In symbols, <math>U = \{U_\alpha : \alpha \in A\}</math> is locally finite if and only if, for any <math>x</math> in <math>X</math>, there exists some neighbourhood <math>V</math> of <math>x</math> such that the set

: <math>\left\{ \alpha \in A : U_{\alpha} \cap V \neq \varnothing \right\}</math>

is finite. A topological space <math>X</math> is now said to be '''paracompact''' if every open cover has a locally finite open refinement.

This definition extends verbatim to locales, with the exception of locally finite: an open cover <math>U</math> of <math>X</math> is locally finite iff the set of opens <math>V</math> that intersect only finitely many opens in <math>U</math> also form a cover of <math>X</math>. Note that an open cover on a topological space is locally finite iff its a locally finite cover of the underlying locale.