## Proof That if the Domain is a T2 Space and a Function is Onto and One to One, and the Inverse Function is Continuous, then the Codomain is a T2 Space

Theorem

Ifis one to one and onto,is continuous andis a T2 space, thenis a T2 space.

Proof

Letandrepresent any two points ofIfis one to one and onto, two distinct points x_1 , x_2 in X exist such thaand

is a Hausdorff space so there are open setsandsuch that

Since f is bijective,

Sinceis continuous the functionmaps open sets into open sets. Henceare open sets and

Henceis a T2 space.