Proof That if the Class of Real Valued Continuous Functions Separates Points of a Space Then That Space is Hausdorff

Theorem

If the class of real valued continuous functions separates points of a space then that space is hausdorff.

Theorem

Letbe distinct points ofand letbe the class of real valued continuous functions onSinceseparates points, a continuous function exists such that

There are open disjoint subsetsandsuch that

Sinceis continuousandare open and disjoint withand

Henceis a Hausdorff or T2 space.