A topological space \(X\) is called compact if every open cover of \(X\) admits a finite subcover.
A subset \(A\subseteq X\) is called compact if it is compact as topological space endowed with the subspace topology .
Remarks
- Every closed subset of a compact space is compact.
- Every compact subset of a Hausdorff space is closed.
- Every compact subset of a metric space is bounded.
- Every finite product of compact spaces is compact.
- Every quotient of a compact space is compact.