reserve x,y,X for set;

theorem
  for T being non empty TopSpace for A being Subset of T holds A is open
iff for x being Point of T st x in A for F being Filter of BoolePoset [#]T st x
  is_a_convergence_point_of F, T holds A in F
proof
  let T be non empty TopSpace, A be Subset of T;
  thus A is open implies for x being Point of T st x in A for F being Filter
of BoolePoset [#]T st x is_a_convergence_point_of F, T holds A in F;
  assume
A1: for x being Point of T st x in A for F being Filter of BoolePoset
  [#]T st x is_a_convergence_point_of F, T holds A in F;
  set x = the Element of A \ Int A;
A2: Int A c= A by TOPS_1:16;
  assume A is not open;
  then not A c= Int A by A2,XBOOLE_0:def 10;
  then
A3: A \ Int A <> {} by XBOOLE_1:37;
  then x in A \ Int A;
  then reconsider x as Point of T;
A4: x in A by A3,XBOOLE_0:def 5;
  x is_a_convergence_point_of NeighborhoodSystem x, T by Th3;
  then A in NeighborhoodSystem x by A1,A4;
  then A is a_neighborhood of x by Th2;
  then x in Int A by CONNSP_2:def 1;
  hence thesis by A3,XBOOLE_0:def 5;
end;
