
theorem Th22:
  for T being Hausdorff non empty TopSpace holds T is sober
proof
  let T be Hausdorff non empty TopSpace;
  let S be irreducible Subset of T;
  consider d be Element of S such that
A1: S = {d} by SUBSET_1:46;
  reconsider d as Point of T;
  take d;
  thus d is_dense_point_of S by A1,Th19;
  let p be Point of T;
  assume p is_dense_point_of S;
  then p in S;
  hence thesis by A1,TARSKI:def 1;
end;
