Theorem

A cartesian product of topological spaces is a topological space.

Proof

Letandbe topological spaces.

Ifandare open thenis open in

From this we conclude that the family of all open setsis a basis for

The identityshows that ifand are open inandrespectively thenandare open in

Since the union of any family of open sets in is open.

Hence the union of two toplogical spaces is a topological space.