Proof That a Function From a Topological Space onto a Cartesian Product is Continuous if and Only if the Component Functions are Continuous

Theorem

Letwhere

is continuous if and only ifandare continuous. Alsois continuous atif and only ifandare continuous at

Proof

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.