Proof That The Class of All Real Valued Continuous Functions on a Completely Regular T1 Space Separates Points
The class of of all real valued continuous functionson a completely regular t1 spaceseparates points.
Suppose we have a completely regular topological spaceand distinct pointsandofSinceis T1 the setis closed. Pointsandare distinct hence
The space (X,T) is completely regular hence a real valued continuous functiononexists such thatand
The function f separates a and b since