Proof That if the Class of Real Valued Continuous Functions Separates Points of a Space Then That Space is Hausdorff
If the class of real valued continuous functions separates points of a space then that space is hausdorff.
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.