
theorem Th27:
  for T being non empty TopSpace holds T is regular iff for p
being Point of T, P being Subset of T st p in Int P ex Q being Subset of T st Q
  is closed & Q c= P & p in Int Q
proof
  let T be non empty TopSpace;
  hereby
    assume
A1: T is regular;
    let p be Point of T, P be Subset of T;
    assume p in Int P;
    then
A2: p in (Int P)``;
    per cases;
    suppose
A3:   P = [#]T;
      take Q = [#]T;
      thus Q is closed;
      thus Q c= P by A3;
      Int Q = Q by TOPS_1:15;
      hence p in Int Q;
    end;
    suppose
   P <> [#]T;
      consider W,V being Subset of T such that
A4:   W is open and
A5:   V is open and
A6:   p in W and
A7:   (Int P)` c= V and
A8:   W misses V by A1,A2;
A9:  Int P c= P by TOPS_1:16;
      take Q = V`;
      thus Q is closed by A5;
      (Int P)` c= Q` by A7;
      then Q c= Int P by SUBSET_1:12;
      hence Q c= P by A9;
      W c= Q by A8,SUBSET_1:23;
      then W c= Int Q by A4,TOPS_1:24;
      hence p in Int Q by A6;
    end;
  end;
  assume
A10: for p being Point of T, P being Subset of T st p in Int P ex Q
  being Subset of T st Q is closed & Q c= P & p in Int Q;
  let p be Point of T, P be Subset of T such that
  P <> {} and
A11: P is closed & p in P`;
  p in Int P` by A11,TOPS_1:23;
  then consider Q being Subset of T such that
A12: Q is closed and
A13: Q c= P` and
A14: p in Int Q by A10;
  reconsider W = Int Q as Subset of T;
  take W, V = Q`;
  thus W is open;
  thus V is open by A12;
  thus p in W by A14;
  P`` c= V by A13,SUBSET_1:12;
  hence P c= V;
  Q misses V by XBOOLE_1:79;
  hence thesis by TOPS_1:16,XBOOLE_1:63;
end;
