Theorem
If
are compact subsets of a topological space
then their union
is also compact.
Proof
Let
be compact subsets ofand let
be an open cover of
so that
Sinceis compact a finite subcover
exists for
Then
andis compact.
Theorem
Ifare compact subsets of a topological spacethen their unionis also compact.
Proof
Letbe compact subsets ofand letbe an open cover ofso thatSinceis compact a finite subcoverexists for
Thenandis compact.