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

theorem
  A is F_sigma implies A` is G_delta
proof
  assume A is F_sigma;
  then consider F being closed countable Subset-Family of T such that
A1: A = union F;
  per cases;
  suppose
A2: F <> {};
    set G = COMPLEMENT F;
A3: G is open by TOPS_2:9;
    (union F)` = meet COMPLEMENT F by A2,TOPS_2:6;
    hence thesis by A1,A3;
  end;
  suppose
    F = {};
    then A` = [#]T by A1,ZFMISC_1:2;
    hence thesis;
  end;
end;
