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
Let
be distinct points of
and let
be the class of real valued continuous functions on
Sinceseparates points, a continuous function
exists such that
There are open disjoint subsets
and
such that
Since
is continuous
and
are open and disjoint with
and
Hence
is a Hausdorff or T2 space.
