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

theorem Th58:
  for A being Subset of X holds A is maximal_discrete iff for x
  being Point of X ex a being Point of X st a in A & A /\ Cl {x} = {a}
proof
  let A be Subset of X;
  thus A is maximal_discrete implies for x being Point of X ex a being Point
  of X st a in A & A /\ Cl {x} = {a}
  proof
    assume
A1: A is maximal_discrete;
    let x be Point of X;
    the carrier of X = union {Cl {a} where a is Point of X : a in A} by A1,Th57
;
    then consider C being set such that
A2: x in C and
A3: C in {Cl {a} where a is Point of X : a in A} by TARSKI:def 4;
    consider a being Point of X such that
A4: C = Cl {a} and
A5: a in A by A3;
A6: A is discrete by A1;
    now
      take a;
      thus a in A by A5;
      Cl {x} = Cl {a} by A2,A4,Th49;
      hence A /\ Cl {x} = {a} by A6,A5,Th36;
    end;
    hence thesis;
  end;
  assume
A7: for x being Point of X ex a being Point of X st a in A & A /\ Cl {x
  } = {a};
A8: for D being Subset of X st D is discrete & A c= D holds A = D
  proof
    let D be Subset of X;
    assume
A9: D is discrete;
    assume
A10: A c= D;
    now
      let x be object;
      assume
A11:  x in D;
      then reconsider y = x as Point of X;
A12:  ex a being Point of X st a in A & A /\ Cl {y} = {a} by A7;
      D /\ Cl {y} = {y} by A9,A11,Th36;
      hence x in A by A10,A12,XBOOLE_1:26,ZFMISC_1:18;
    end;
    then D c= A by TARSKI:def 3;
    hence thesis by A10;
  end;
  for x being Point of X st x in A holds A /\ Cl {x} = {x}
  proof
    let x be Point of X;
A13: {x} c= Cl {x} by PRE_TOPC:18;
    assume x in A;
    then
A14: {x} c= A by ZFMISC_1:31;
    ex a being Point of X st a in A & A /\ Cl {x} = {a} by A7;
    hence thesis by A14,A13,XBOOLE_1:19,ZFMISC_1:18;
  end;
  then A is discrete by Th52;
  hence A is maximal_discrete by A8;
end;
