## Proof That the Graph of a Continuous Function on a Set X is Homeomorphic to X

Theorem

If a graphis continuous then the graph ofis homeomorphic to

Proof

Consider the mappingis a homeomorphism ofsince it is one to one and onto.

Also ifis a T2 space thenis closed inTo show this consider the mappingthenhence

Sinceis continuous andis closed,is closed.