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

theorem Th39:
  for X being non empty set, x0 being Element of X for A being
  proper Subset of x0-PointClTop(X) holds A is open iff not x0 in A
proof
  let X be non empty set;
  let x0 be Element of X;
  let A be proper Subset of x0-PointClTop(X);
  A is open iff A` is closed by TOPS_1:4;
  then
A1: A is open iff x0 in A` by Th38;
  x0 is Element of x0-PointClTop X by Def7;
  hence thesis by A1,XBOOLE_0:def 5;
end;
