Compact Metrizable Spaces

Abstract

A topological space,X is compact if every open cover of X has a finite subcover, i.e., if (U _i ) _i∈I is a family of open sets and X $X = \bigcup\nolimits_{i \in I} {{U_i}}$, then there is finite ${I_0} \subseteq I$ such that $X = \bigcup\nolimits_{i \in {I_0}} {{U_i}}$.This is equivalent to saying that every family of closed subsets of X with the finite intersection property (i.e., one for which every finite subfamily has nonempty intersection) has nonempty intersection.