reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;

theorem Th48:
  for A being Subset of X holds Cl A = union {Cl {a} where a is
  Point of X : a in A}
proof
  let A be Subset of X;
  set F = {Cl {a} where a is Point of X : a in A};
  now
    let C be object;
    assume C in F;
    then ex a being Point of X st C = Cl {a} & a in A;
    hence C in bool the carrier of X;
  end;
  then reconsider F as Subset-Family of X by TARSKI:def 3;
  reconsider F as Subset-Family of X;
  now
    let x be object;
    assume
A1: x in A;
    then reconsider a = x as Point of X;
    Cl {a} in F by A1;
    then
A2: Cl {a} c= union F by ZFMISC_1:74;
A3: {a} c= Cl {a} by PRE_TOPC:18;
    a in {a} by TARSKI:def 1;
    then a in Cl {a} by A3;
    hence x in union F by A2;
  end;
  then
A4: A c= union F by TARSKI:def 3;
  now
    let C be set;
    assume C in F;
    then consider a being Point of X such that
A5: C = Cl {a} and
A6: a in A;
    {a} c= A by A6,ZFMISC_1:31;
    hence C c= Cl A by A5,PRE_TOPC:19;
  end;
  then
A7: union F c= Cl A by ZFMISC_1:76;
  now
    let G be Subset of X;
    assume G in F;
    then ex a being Point of X st G = Cl {a} & a in A;
    hence G is open by TDLAT_3:22;
  end;
  then F is open by TOPS_2:def 1;
  then union F is open by TOPS_2:19;
  then union F is closed by TDLAT_3:21;
  then Cl A c= union F by A4,TOPS_1:5;
  hence thesis by A7;
end;
