
theorem Th22:
  for T being non empty TopSpace, A being Subset of T, p being
Point of T holds p is_isolated_in A iff ex G being open Subset of T st G /\ A =
  {p}
proof
  let T be non empty TopSpace, A be Subset of T, p be Point of T;
  hereby
    assume
A1: p is_isolated_in A;
    then not p is_an_accumulation_point_of A;
    then consider U being open Subset of T such that
A2: p in U and
A3: not ex q being Point of T st q <> p & q in A & q in U by Th21;
    take U;
A4: p in A by A1;
    U /\ A = {p}
    proof
      thus U /\ A c= {p}
      proof
        let x be object;
        assume x in U /\ A;
        then x in U & x in A by XBOOLE_0:def 4;
        then x = p by A3;
        hence thesis by TARSKI:def 1;
      end;
      let x be object;
      assume x in {p};
      then x = p by TARSKI:def 1;
      hence thesis by A4,A2,XBOOLE_0:def 4;
    end;
    hence U /\ A = {p};
  end;
  given G being open Subset of T such that
A5: G /\ A = {p};
A6: p in G /\ A by A5,TARSKI:def 1;
  ex U being open Subset of T st p in U & not ex q being Point of T st q
  <> p & q in A & q in U
  proof
    take G;
    for q being Point of T holds q = p or not q in A or not q in G
    proof
      given q being Point of T such that
A7:   q <> p and
A8:   q in A & q in G;
      q in A /\ G by A8,XBOOLE_0:def 4;
      hence thesis by A5,A7,TARSKI:def 1;
    end;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
  then
A9: not p is_an_accumulation_point_of A by Th21;
  p in A by A6,XBOOLE_0:def 4;
  hence thesis by A9;
end;
