Proof That a Function From a Topological Space onto a Cartesian Product is Continuous if and Only if the Component Functions are Continuous
is continuous if and only ifandare continuous. Alsois continuous atif and only ifandare continuous at
Suppose thatis continuous atSinceis continuous at
Suppose now thatandare continuous atLetbelong to a subbase ofand let
Setand we obtainhence
Butis continuous at t_0 thus
The proof foris almost identical. Henceis continuous if and only ifandare.