
theorem
  for X being non empty TopSpace st X is non compact holds incl(X,
  One-Point_Compactification X) is compactification
proof
  let X be non empty TopSpace;
  set h = incl(X,One-Point_Compactification X);
  set D = {U \/ {[#]X} where U is Subset of X: U is open & U` is compact};
  assume X is non compact;
  then
A1: ex X9 being Subset of One-Point_Compactification(X) st X9 = [#]X & X9
  is dense by Th7;
A2: [#]X c= [#](One-Point_Compactification X) by Th4;
  then reconsider Xy = [#]X as Subset of One-Point_Compactification(X);
A3: [#]((One-Point_Compactification X) | Xy) = Xy by PRE_TOPC:def 5;
A4: the topology of One-Point_Compactification(X) = (the topology of X) \/ D
  by Def9;
  the topology of (One-Point_Compactification X) | Xy = the topology of X
  proof
    thus the topology of (One-Point_Compactification X) |Xy c= the topology of
    X
    proof
      let x be object;
      assume
A5:   x in the topology of (One-Point_Compactification X) | Xy;
      then reconsider P = x as Subset of (One-Point_Compactification X) | Xy;
      consider Q being Subset of One-Point_Compactification(X) such that
A6:   Q in the topology of One-Point_Compactification(X) and
A7:   P = Q /\ [#]((One-Point_Compactification X) | Xy) by A5,PRE_TOPC:def 4;
      per cases by A4,A6,XBOOLE_0:def 3;
      suppose
        Q in the topology of X;
        hence thesis by A3,A7,XBOOLE_1:28;
      end;
      suppose
        Q in D;
        then consider U being Subset of X such that
A8:     Q = U \/ {[#]X} and
A9:     U is open and
        U` is compact;
        not [#]X in [#]X;
        then {[#]X} misses [#]X by ZFMISC_1:50;
        then {[#]X} /\ [#]X = {};
        then P = (U /\ [#]X) \/ {} by A3,A7,A8,XBOOLE_1:23
          .= U by XBOOLE_1:28;
        hence thesis by A9;
      end;
    end;
    let x be object;
    assume
A10: x in the topology of X;
    then reconsider
    P = x as Subset of (One-Point_Compactification X) | Xy by A3;
    reconsider Q = P as Subset of One-Point_Compactification(X) by A3,
XBOOLE_1:1;
A11: P = Q /\ [#]((One-Point_Compactification X) | Xy) by XBOOLE_1:28;
    Q in the topology of One-Point_Compactification(X) by A4,A10,XBOOLE_0:def 3
;
    hence thesis by A11,PRE_TOPC:def 4;
  end;
  hence h is embedding by A2,Th3;
  thus One-Point_Compactification(X) is compact;
  thus thesis by A1,Th2;
end;
