A Topology on R That is T2 But Not T3

Define the family of open sets onas the set of all open intervals containingtogether with the family of open sets of the formWith this definitionis T2 but not T3.

To show thatis T2 ,Ifthen take

andare open sets and

Ifthen take

Henceis a T2 or Hausdorff space.

Now letandF is closed.

LetNo open setexists such thatand

Henceis not regular, so not T3.