Proof That a Function From a Topological Space onto a Cartesian Product is Continuous if and Only if the Component Functions are Continuous
Theorem
Let
where
is continuous if and only if
and
are continuous. Alsois continuous at
if and only ifandare continuous at
Proof
Suppose thatis continuous at
Since
is continuous at
Suppose now thatandare continuous atLet
belong to a subbase of
and let
Set
and we obtain
hence
Butis continuous at t_0 thus
The proof foris almost identical. Henceis continuous if and only ifandare.
