
theorem
  for X being non empty TopSpace holds X is Hausdorff locally-compact
  iff One-Point_Compactification(X) is Hausdorff
proof
  let X be non empty TopSpace;
  set D = {U \/ {[#]X} where U is Subset of X: U is open & U` is compact};
A1: not [#]X in [#]X;
A2: [#]One-Point_Compactification(X) = succ([#]X) by Def9;
  [#]X in {[#]X} by TARSKI:def 1;
  then reconsider q = [#]X as Point of One-Point_Compactification(X) by A2,
XBOOLE_0:def 3;
A3: the topology of One-Point_Compactification(X) = (the topology of X) \/ D
  by Def9;
A4: [#]X c= [#]One-Point_Compactification(X) by Th4;
  thus X is Hausdorff locally-compact implies One-Point_Compactification(X) is
  Hausdorff
  proof
    assume that
A5: X is Hausdorff and
A6: X is locally-compact;
    let p, q be Point of One-Point_Compactification(X) such that
A7: p <> q;
    per cases by A2,XBOOLE_0:def 3;
    suppose
      p in [#]X & q in [#]X;
      then consider W, V being Subset of X such that
A8:   W is open and
A9:   V is open and
A10:  p in W and
A11:  q in V and
A12:  W misses V by A5,A7;
      reconsider W,V as Subset of One-Point_Compactification(X) by A4,
XBOOLE_1:1;
      V in the topology of X by A9;
      then
A13:  V in the topology of One-Point_Compactification(X) by A3,XBOOLE_0:def 3;
      take W,V;
      W in the topology of X by A8;
      then W in the topology of One-Point_Compactification(X) by A3,
XBOOLE_0:def 3;
      hence W is open & V is open by A13;
      thus thesis by A10,A11,A12;
    end;
    suppose that
A14:  p in [#]X and
A15:  q in {[#]X};
      reconsider px = p as Point of X by A14;
      consider P being a_neighborhood of px such that
A16:  P is compact by A6;
      [#]X c= [#]One-Point_Compactification(X) by A2,XBOOLE_1:7;
      then reconsider W = Int P as Subset of One-Point_Compactification(X) by
XBOOLE_1:1;
      P`` = P;
      then P` \/ {[#]X} in D by A5,A16;
      then
A17:  P` \/ {[#]X} in the topology of One-Point_Compactification(X) by A3,
XBOOLE_0:def 3;
      then reconsider
      V = P`\/{[#]X} as Subset of One-Point_Compactification(X );
      take W,V;
      W in the topology of X by PRE_TOPC:def 2;
      then W in the topology of One-Point_Compactification(X) by A3,
XBOOLE_0:def 3;
      hence W is open;
      thus V is open by A17;
      thus p in W by CONNSP_2:def 1;
      thus q in V by A15,XBOOLE_0:def 3;
      not [#]X in [#]X;
      then not [#]X in Int P;
      then
A18:  Int P misses {[#]X} by ZFMISC_1:50;
      Int P c= P by TOPS_1:16;
      then Int P misses P` by SUBSET_1:24;
      hence thesis by A18,XBOOLE_1:70;
    end;
    suppose that
A19:  q in [#]X and
A20:  p in {[#]X};
      reconsider qx = q as Point of X by A19;
      consider Q being a_neighborhood of qx such that
A21:  Q is compact by A6;
      [#]X c= [#]One-Point_Compactification(X) by Th4;
      then reconsider W = Int Q as Subset of One-Point_Compactification(X) by
XBOOLE_1:1;
      Q`` = Q;
      then Q` \/ {[#]X} in D by A5,A21;
      then
A22:  Q` \/ {[#]X} in the topology of One-Point_Compactification(X) by A3,
XBOOLE_0:def 3;
      then reconsider
      V = Q`\/{[#]X} as Subset of One-Point_Compactification(X );
      take V,W;
      thus V is open by A22;
      W in the topology of X by PRE_TOPC:def 2;
      then W in the topology of One-Point_Compactification(X) by A3,
XBOOLE_0:def 3;
      hence W is open;
      thus p in V by A20,XBOOLE_0:def 3;
      thus q in W by CONNSP_2:def 1;
      not [#]X in [#]X;
      then not [#]X in Int Q;
      then
A23:  Int Q misses {[#]X} by ZFMISC_1:50;
      Int Q c= Q by TOPS_1:16;
      then Int Q misses Q` by SUBSET_1:24;
      hence thesis by A23,XBOOLE_1:70;
    end;
    suppose
A24:  p in {[#]X} & q in {[#]X};
      then p = [#]X by TARSKI:def 1;
      hence thesis by A7,A24,TARSKI:def 1;
    end;
  end;
A25: X is SubSpace of One-Point_Compactification(X) by Th5;
  assume
A26: One-Point_Compactification(X) is Hausdorff;
  hence X is Hausdorff by A25;
  let x be Point of X;
  reconsider p = x as Point of One-Point_Compactification(X) by A4;
  not [#]X in [#]X;
  then p <> q;
  then consider V, U being Subset of One-Point_Compactification(X) such that
A27: V is open and
A28: U is open and
A29: p in V and
A30: q in U and
A31: V misses U by A26;
A32: now
    assume U in the topology of X;
    then q in [#]X by A30;
    hence contradiction;
  end;
  U in the topology of One-Point_Compactification(X) by A28;
  then U in D by A3,A32,XBOOLE_0:def 3;
  then consider W being Subset of X such that
A33: U = W \/ {[#]X} and
  W is open and
A34: W` is compact;
A35: [#]X \ U = ([#]X \ W) /\ ([#]X \ {[#]X}) by A33,XBOOLE_1:53
    .= ([#]X \ W) /\ [#]X by A1,ZFMISC_1:57
    .= [#]X \ W by XBOOLE_1:28;
A36: [#]X in {[#]X} by TARSKI:def 1;
  then
A37: [#]X in U by A33,XBOOLE_0:def 3;
A38: [#]One-Point_Compactification(X) \ U = ([#]X \ U) \/ ({[#]X} \ U) by A2,
XBOOLE_1:42
    .= ([#]X \ W ) \/ {} by A37,A35,ZFMISC_1:60
    .= W`;
A39: V c= U` by A31,SUBSET_1:23;
  then
A40: V c= [#]X by A38,XBOOLE_1:1;
A41: now
    assume V in D;
    then
    ex S being Subset of X st V = S \/ {[#]X} & S is open & S` is compact;
    then [#]X in V by A36,XBOOLE_0:def 3;
    then [#]X in [#]X by A40;
    hence contradiction;
  end;
  V in the topology of One-Point_Compactification(X) by A27;
  then V in the topology of X by A3,A41,XBOOLE_0:def 3;
  then reconsider Vx = V as open Subset of X by PRE_TOPC:def 2;
  set K = Cl Vx;
A42: Int Vx c= Int K by PRE_TOPC:18,TOPS_1:19;
  x in Int Vx by A29,TOPS_1:23;
  then reconsider K as a_neighborhood of x by A42,CONNSP_2:def 1;
  take K;
  U` /\ [#]X = W` by A38,XBOOLE_1:28;
  then W` is closed by A25,A28,PRE_TOPC:13;
  hence thesis by A34,A39,A38,COMPTS_1:9,TOPS_1:5;
end;
