Non Intersection Property for Compact Sets
A topological spaceis compact if and only if every setof closed subsets of having the non intersection property, has non zero intersection.
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