
theorem Th21:
  for T being non empty TopSpace, A being Subset of T, p being
Point of T holds p is_an_accumulation_point_of A iff for U being open Subset of
  T st p in U ex q being Point of T st q <> p & q in A & q in U
proof
  let T be non empty TopSpace, A be Subset of T, p be Point of T;
  hereby
    assume p is_an_accumulation_point_of A;
    then
A1: p in Der A by Th16;
    let U be open Subset of T;
    assume p in U;
    then consider q being Point of T such that
A2: q in A /\ U & p <> q by A1,Th17;
    take q;
    thus q <> p & q in A & q in U by A2,XBOOLE_0:def 4;
  end;
  assume
A3: for U being open Subset of T st p in U ex q being Point of T st q <>
  p & q in A & q in U;
  for U being open Subset of T st p in U ex y being Point of T st y in A
  /\ U & p <> y
  proof
    let U be open Subset of T;
    assume p in U;
    then consider q being Point of T such that
A4: q <> p & q in A & q in U by A3;
    take q;
    thus thesis by A4,XBOOLE_0:def 4;
  end;
  then p in Der A by Th17;
  hence thesis by Th16;
end;
