Proof That Every T2 Space is a T1 Space
A topological space is a T2 or Hausdorff /separated space if, for each pair of distinct points
and 
disjoint open sets
and
exist such that
A metric space is obviously T2.
Suppose now
is a T2 space. Let
There is an open set A_x containing x with a notin A_x .
Hence
and
is an open set since it is a union of a family of open sets
Hence
is closed and
is a T1 space.
