
theorem
  for T being TopSpace ex F being Subset-Family of T st F = {the carrier
  of T} & F is Cover of T & F is open
proof
  let T be TopSpace;
  set F = {the carrier of T};
  F c= bool the carrier of T
  proof
    let a be object;
    assume a in F;
    then a = the carrier of T by TARSKI:def 1;
    hence thesis by ZFMISC_1:def 1;
  end;
  then reconsider F as Subset-Family of T;
  take F;
  thus F = {the carrier of T};
  the carrier of T c= union F by ZFMISC_1:25;
  hence F is Cover of T by SETFAM_1:def 11;
  let P be Subset of T;
  [#]T = the carrier of T;
  hence thesis by TARSKI:def 1;
end;
