Paracompact Space - Paracompactness

Paracompactness

A cover of a set X is a collection of subsets of X whose union contains X. In symbols, if U = {U_α : α in A} is an indexed family of subsets of X, then U is a cover of X if

A cover of a topological space X is open if all its members are open sets. 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 = {V_β : β in B} is a refinement of the cover U = {U_α : α in A} if and only if, for any V_β in V, there exists some U_α in U such that V_β is contained in U_α.

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 = {U_α : 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

is finite.