
theorem Th27:
  for T being non empty TopSpace for F being ultra Filter of
  BoolePoset the carrier of T for p being set holds p is_a_cluster_point_of F,T
  iff p is_a_convergence_point_of F,T
proof
  let T be non empty TopSpace;
  set L = BoolePoset the carrier of T;
  let F be ultra Filter of L;
  let p be set;
  thus p is_a_cluster_point_of F,T implies p is_a_convergence_point_of F,T
  proof
    assume
A1: for A being Subset of T st A is open & p in A for B being set st B
    in F holds A meets B;
    let A be Subset of T;
    assume that
A2: A is open & p in A and
A3: not A in F;
    F is prime by Th22;
    then (the carrier of T)\A in F by A3,Th21;
    then A meets (the carrier of T)\A by A1,A2;
    hence contradiction by XBOOLE_1:79;
  end;
  assume
A4: for A being Subset of T st A is open & p in A holds A in F;
  let A be Subset of T;
  assume A is open & p in A;
  then
A5: A in F by A4;
  Bottom L = {} by YELLOW_1:18;
  then
A6: not {} in F by Th4;
  let B be set;
  assume
A7: B in F;
  then reconsider a = A, b = B as Element of L by A5;
  a"/\"b = A /\ B by YELLOW_1:17;
  then A /\ B in F by A5,A7,WAYBEL_0:41;
  hence thesis by A6;
end;
