
theorem
  for T being non empty TopSpace st T is normal & T is T_1 holds for A
being open Subset of T st A <> {} holds ex B being Subset of T st B <> {} & Cl(
  B) c= A
proof
  let T be non empty TopSpace;
  assume that
A1: T is normal and
A2: T is T_1;
  let A be open Subset of T;
  assume
A3: A <> {};
  now
    per cases;
    case
   A <> [#](T);
      reconsider V = [#](T) \ A as Subset of T;
      consider x being object such that
A4:   x in A by A3,XBOOLE_0:def 1;
      A = [#](T) \ V by PRE_TOPC:3;
      then
A5:   V is closed;
      reconsider x as Point of T by A4;
      consider W being set such that
A6:   W = {x};
      reconsider W as Subset of T by A6;
A7:   W misses V
      proof
        assume W meets V;
        then consider z being object such that
A8:    z in W /\ V by XBOOLE_0:4;
        z in W by A8,XBOOLE_0:def 4;
        then
A9:    z in A by A4,A6,TARSKI:def 1;
        z in V by A8,XBOOLE_0:def 4;
        hence thesis by A9,XBOOLE_0:def 5;
      end;
      W is closed by A2,A6,Th19;
      then consider B,Q being Subset of T such that
      B is open and
A10:  Q is open and
A11:  W c= B and
A12:  V c= Q and
A13:  B misses Q by A1,A5,A7;
      take B;
      B <> {} & Cl(B) c= A
      proof
        B c= Q` by A13,SUBSET_1:23;
        then Cl(B) c= Q` by A10,TOPS_1:5;
        then Cl(B) misses Q by SUBSET_1:23;
        then
A14:    V misses Cl(B) by A12,XBOOLE_1:63;
        A`` = A;
        hence thesis by A6,A11,A14,SUBSET_1:23;
      end;
      hence thesis;
    end;
    case
A15:  A = [#](T);
      consider B being Subset of T such that
A16:  B = [#](T);
      take B;
      Cl(B) c= A by A15;
      hence thesis by A16;
    end;
  end;
  hence thesis;
end;
