Suppose we have a set
with the topology
The closed sets are
Not every singleton set of
is closed since for example
hence the topology is not T1.
The topology is not regular.
is closed and
so the only open set containingis which also contains
Hence
is not regular.
The topological space is normal. If
and
are disjoint closed subsets ofthen one of them must be
Suppose
then
and
Henceis normal.