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

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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

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