Theorem
A cartesian product of topological spaces is a topological space.
Proof
Let
and
be topological spaces.
If
and
are open then
is open in
From this we conclude that the family of all open setsis a basis for
The identity
shows that if
and
are open in
and
respectively then
and
are 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.