
theorem Th23:
  for T being non empty TopSpace holds T is T_0 iff for p,q being
  Point of T st Cl{p} = Cl{q} holds p = q
proof
  let T be non empty TopSpace;
  thus T is T_0 implies for p,q being Point of T st Cl{p} = Cl{q} holds p = q
  by TSP_1:def 5;
  assume for p,q being Point of T st Cl{p} = Cl{q} holds p = q;
  then for p,q being Point of T st p <> q holds Cl{p} <> Cl{q};
  hence thesis by TSP_1:def 5;
end;
