Proof That a Non Constant Continuous Function Exists on a T4 Space Containing More Than One Point


Ifis a T4 space containing more than one point, then there exists a non constant continuous function


Supposeare distinct points. Sinceis T4 it must be normal and T1.

For any T1 space


so that any singleton set is closed. Sinceandare distinctandare disjoint.

Therefore, sinceis normal a continuous functionexists such that and

Henceis a continuous non constant function frominto