Define the family of open sets on
as the set of all open intervals containing
together with the family of open sets of the form
With this definition
is T2 but not T3.
To show thatis T2 ,
If
then take
and
are open sets and
If
then take
Henceis a T2 or Hausdorff space.
Now letand
F is closed.
Let
No open setexists such that
and
Henceis not regular, so not T3.