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
Henceis a T2 or Hausdorff space.
Now letandF is closed.
LetNo open setexists such thatand
Henceis not regular, so not T3.