Theorem
If a topological space
is a T1 space then for any pair of distinct points
open sets
exist with
Proof
Supposeis a T1 space. For any
is closed. Let
be distinct points of
and
are closed so
and
are open.
Also
and
and
Conversely suppose
is open. Let
then
and an open set
exists such that
Hence
and 
Since allare openis open andis closed so