Theorem
If a topological spaceis 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.
Alsoand
and
Conversely supposeis open. Let
then
and an open set
exists such that
Hence
and
Since allare open
is open and
is closed so