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 in X holds {x} is closed Subset of x0
  -PointClTop(X) iff x = x0
proof
  let X,x0,x be set;
  assume
A1: x0 in X;
  hereby
    assume {x} is closed Subset of x0-PointClTop(X);
    then x0 in {x} by A1,Th38;
    hence x = x0 by TARSKI:def 1;
  end;
  assume
A2: x = x0;
  then
A3: x0 in {x} by ZFMISC_1:31;
  {x} c= X by A2,A1,ZFMISC_1:31;
  hence thesis by A3,Def7,Th38;
end;
