Proof That a Cartesian Product of Topological Spaces is a Topological Space
A cartesian product of topological spaces is a topological space.
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.