Theorem
The class of of all real valued continuous functions
on a completely regular t1 space
separates points.
Proof
Suppose we have a completely regular topological spaceand distinct points
and
of
Sinceis T1 the set
is closed. Pointsandare distinct hence
The space (X,T) is completely regular hence a real valued continuous function
on
exists such that
and
The function f separates a and b since