
theorem Th7:
  for X being non empty TopSpace holds X is non compact iff ex X9
  being Subset of One-Point_Compactification(X) st X9 = [#]X & X9 is dense
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;
A3: [#]X in {[#]X} by TARSKI:def 1;
  then
A4: [#]X in [#]One-Point_Compactification(X) by A2,XBOOLE_0:def 3;
A5: the topology of One-Point_Compactification(X) = (the topology of X) \/ D
  by Def9;
  thus X is non compact implies ex X9 being Subset of
  One-Point_Compactification(X) st X9 = [#]X & X9 is dense
  proof
    assume X is non compact;
    then
A6: [#]X is non compact;
    [#]X c= [#]One-Point_Compactification(X) by Th4;
    then reconsider X9 = [#]X as Subset of One-Point_Compactification(X);
    take X9;
    thus X9 = [#]X;
    thus Cl(X9) c= [#]One-Point_Compactification(X);
A7: [#]X c= Cl(X9) by PRE_TOPC:18;
A8: now
      reconsider Xe = [#]X as Element of One-Point_Compactification(X) by A3,A2
,XBOOLE_0:def 3;
      assume
A9:   not [#]X in Cl(X9);
      reconsider XX = {Xe} as Subset of One-Point_Compactification(X);
A10:  now
        assume XX in the topology of X;
        then [#]X in [#]X by A3;
        hence contradiction;
      end;
      [#]One-Point_Compactification(X) \ Cl(X9) = ([#]X \ Cl(X9)) \/ ({
      [#]X} \ Cl(X9)) by A2,XBOOLE_1:42
        .= {} \/ ({[#]X} \ Cl(X9)) by A7,XBOOLE_1:37
        .= XX by A9,ZFMISC_1:59;
      then XX is open by PRE_TOPC:def 3;
      then XX in the topology of One-Point_Compactification(X);
      then XX in D by A5,A10,XBOOLE_0:def 3;
      then consider U being Subset of X such that
A11:  XX = U \/ {[#]X} and
      U is open and
A12:  U` is compact;
      now
        assume U = XX;
        then [#]X in [#]X by A3;
        hence contradiction;
      end;
      then U = {}X by A11,ZFMISC_1:37;
      hence contradiction by A6,A12;
    end;
    [#]One-Point_Compactification(X) c= Cl(X9) \/ {[#]X} by A2,PRE_TOPC:18
,XBOOLE_1:9;
    hence thesis by A8,ZFMISC_1:40;
  end;
  given X9 being Subset of One-Point_Compactification(X) such that
A13: X9 = [#]X and
A14: X9 is dense;
A15: Cl X9= [#]One-Point_Compactification(X) by A14;
  assume X is compact;
  then [#]X is compact;
  then ({}X)` is compact;
  then {}X \/ {[#]X} in D;
  then
A16: {[#]X} in the topology of One-Point_Compactification(X) by A5,
XBOOLE_0:def 3;
  then reconsider X1 = {[#]X} as Subset of One-Point_Compactification(X);
  X1 is open by A16;
  then
A17: (Cl X1`) = X1` by PRE_TOPC:22;
  X1` = [#]X \ X1 by A2,XBOOLE_1:40
    .= [#]X by A1,ZFMISC_1:57;
  hence contradiction by A13,A15,A17,A4;
end;
