
theorem
  for T being non empty TopSpace holds for A being Subset of T holds (
  for x being Point of T st x in A holds A is Nbhd of x,T) implies A is open
proof
  let T be non empty TopSpace;
  let A be Subset of T;
  assume
A1: for x being Point of T st x in A holds A is Nbhd of x,T;
  for x being Point of T holds x in A implies ex B being Nbhd of x,T st B c= A
  proof
    let x be Point of T;
    assume
A2: x in A;
    ex B being Nbhd of x,T st B c= A
    proof
      A is Nbhd of x,T by A1,A2;
      then consider B being Nbhd of x,T such that
A3:   B = A;
      take B;
      thus thesis by A3;
    end;
    hence thesis;
  end;
  hence thesis by Th17;
end;
