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 pointsand disjoint open setsandexist such that

A metric space is obviously T2.

Suppose nowis a T2 space. LetThere is an open set A_x containing x with a notin A_x .

Henceandis an open set since it is a union of a family of open sets

Henceis closed andis a T1 space.