
theorem
  for T being non empty TopSpace holds T is regular iff for A being open
Subset of T for p being Point of T st p in A holds ex B being open Subset of T
  st p in B & Cl(B) c= A
proof
  let T be non empty TopSpace;
  thus T is regular implies for A being open Subset of T for p being Point of
  T st p in A holds ex B being open Subset of T st p in B & Cl(B) c= A
  proof
    assume
A1: T is regular;
    thus for A being open Subset of T for p being Point of T st p in A holds
    ex B being open Subset of T st p in B & Cl(B) c= A
    proof
      let A be open Subset of T;
      let p be Point of T;
      assume
A2:   p in A;
      then
A3:   p in A``;
      thus ex B being open Subset of T st p in B & Cl(B) c= A
      proof
        reconsider P = A` as Subset of T;
        now
          per cases;
          case
            P <> {};
            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:         P c= V and
A8:         W misses V by A1,A3;
            W /\ V = {} by A8;
            then V /\ Cl(W) c= Cl({} T) by A5,TOPS_1:13;
            then V /\ Cl(W) c= {} by PRE_TOPC:22;
            then V /\ Cl(W) = {};
            then V misses Cl(W);
            then
A9:         P misses Cl(W) by A7,XBOOLE_1:63;
            take W;
            A`` = A;
            then Cl(W) c= A by A9,SUBSET_1:23;
            hence thesis by A4,A6;
          end;
          case
A10:        P = {};
            take A;
            A = [#](T) by A10,PRE_TOPC:4;
            then Cl(A) c= A;
            hence thesis by A2;
          end;
        end;
        hence thesis;
      end;
    end;
  end;
  assume
A11: for A being open Subset of T for p being Point of T st p in A holds
  ex B being open Subset of T st p in B & Cl(B) c= A;
  for p being Point of T for P being Subset of T st P <> {} & P is closed
& p in P` ex W,V being Subset of T st W is open & V is open & p in W & P c= V &
  W misses V
  proof
    let p be Point of T;
    let P be Subset of T;
    assume that
    P <> {} and
A12: P is closed & p in P`;
    thus ex W,V being Subset of T st W is open & V is open & p in W & P c= V &
    W misses V
    proof
      consider A being Subset of T such that
A13:  A = P`;
      consider B being open Subset of T such that
A14:  p in B & Cl(B) c= A by A11,A12,A13;
      consider C being Subset of T such that
A15:  C = [#](T) \ Cl(B);
      reconsider B,C as Subset of T;
      Cl(B) misses C by A15,XBOOLE_1:79;
      then
A16:  B misses C by PRE_TOPC:18,XBOOLE_1:63;
      take B;
      take C;
      (Cl(B))` is open & P`` = P;
      hence thesis by A13,A14,A15,A16,XBOOLE_1:34;
    end;
  end;
  hence T is regular by COMPTS_1:def 2;
end;
