reserve a,b,c for set;
reserve r for Real,
  X for set,
  n for Element of NAT;

theorem
  for X,x0,x being set st {x0} c< X holds {x} is open Subset of x0
  -PointClTop(X) iff x in X & x <> x0
proof
  let X,x0,x be set;
  assume
A1: {x0} c< X;
  then reconsider Y = X as non empty set;
  reconsider A = {x0} as Subset of Y by A1;
A2: x0 in A by TARSKI:def 1;
  reconsider A as proper Subset of Y by A1,SUBSET_1:def 6;
A3: the carrier of x0-PointClTop(X) = X by Def7;
  hereby
    assume
A4: {x} is open Subset of x0-PointClTop(X);
    hence x in X by A3,ZFMISC_1:31;
    assume x = x0;
    then not x0 in {x0} or A is non proper Subset of x0-PointClTop(X) by A4
,Th39;
    hence contradiction by A3,TARSKI:def 1;
  end;
  assume that
A5: x in X and
A6: x <> x0;
A7: not x0 in {x} by A6,TARSKI:def 1;
  x0 in Y by A2;
  then {x} is proper Subset of x0-PointClTop(Y) by A7,A5,A3,SUBSET_1:def 6
,ZFMISC_1:31;
  hence thesis by A7,A2,Th39;
end;
