reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;
reserve x for Point of T;

theorem
  A^0 = Cl (A^0)
proof
  thus A^0 c= Cl (A^0) by PRE_TOPC:18;
  let x be object;
  assume
A1: x in Cl (A^0);
  then reconsider p = x as Point of T;
  for N being a_neighborhood of p holds N /\ A is not countable
  proof
    let N be a_neighborhood of p;
    consider N1 being Subset of T such that
A2: N1 is open and
A3: N1 c= N and
A4: p in N1 by CONNSP_2:6;
    A^0 meets N1 by A1,A2,A4,PRE_TOPC:24;
    then consider y being object such that
A5: y in A^0 and
A6: y in N1 by XBOOLE_0:3;
    reconsider y9 = y as Point of T by A5;
    reconsider N1 as a_neighborhood of y9 by A2,A6,CONNSP_2:6;
A7: N1 /\ A c= N /\ A by A3,XBOOLE_1:26;
    y9 is_a_condensation_point_of A by A5,Def10;
    hence thesis by A7;
  end;
  then p is_a_condensation_point_of A;
  hence thesis by Def10;
end;
