A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X. A subset A of a topological space X is compact if it is compact as a topological space with the relative topology (i.e., every family of open sets of X whose union contains A has a finite subfamily whose union contains A).

compact set | Heine-Borel theorem | paracompact space | topological space