Theorem

A topological spaceis compact if and only if every setof closed subsets of having the non intersection property, has non zero intersection.

Proof

A familyof sets is said to have the finite intersection property if every finite collectionhas a non empty intersection, so that

Ifis compact, then for every familyof closed subsets ofwiththen contains a finite subsetwithfor some

Now let a and b represent logical statements satisfying(1)

Take a and b as the statementsandfor somerespectively.

is the statementandis the statementfor all

Then (1) implies for allwe have