Proof That The Identity Function is Continuous if and only if the Destination Topology is Finer Than The Source Topology

Theorem

The identity functionis continuous if and only ifwhere

Proof

Suppose thatis continuous then

But hence

Now suppose thatis the identity function so thatand suppose

Let A in T_1 thenbecause T_1 in T hence f is continuous.

This theorem may be applied recursively so that if f,g,h are identity functions