
theorem
  for T being non empty TopSpace holds T is Baire iff for F being
Subset-Family of T st F is countable & for S being Subset of T st S in F holds
  S is nowhere_dense holds union F is boundary
proof
  let T be non empty TopSpace;
  hereby
    assume
A1: T is Baire;
    let F be Subset-Family of T such that
A2: F is countable and
A3: for S being Subset of T st S in F holds S is nowhere_dense;
    reconsider G = COMPLEMENT F as Subset-Family of T;
A4: for S being Subset of T st S in G holds S is everywhere_dense
    proof
      let S be Subset of T;
      assume S in G;
      then S` in F by SETFAM_1:def 7;
      then S` is nowhere_dense by A3;
      hence thesis by TOPS_3:39;
    end;
    G is countable by A2,Th3,Th4;
    then ex I being Subset of T st I = Intersect G & I is dense by A1,A4;
    then (union F)` is dense by Th6;
    hence union F is boundary by TOPS_1:def 4;
  end;
  assume
A5: for F being Subset-Family of T st F is countable & for S being
  Subset of T st S in F holds S is nowhere_dense holds union F is boundary;
  let F be Subset-Family of T such that
A6: F is countable and
A7: for S being Subset of T st S in F holds S is everywhere_dense;
  reconsider I = Intersect F as Subset of T;
  take I;
  thus I = Intersect F;
  reconsider G = COMPLEMENT F as Subset-Family of T;
A8: for S being Subset of T st S in G holds S is nowhere_dense
  proof
    let S be Subset of T;
    assume S in G;
    then S` in F by SETFAM_1:def 7;
    then S` is everywhere_dense by A7;
    hence thesis by TOPS_3:40;
  end;
  G is countable by A6,Th3,Th4;
  then union G is boundary by A5,A8;
  then (Intersect F)` is boundary by Th7;
  hence thesis by TOPS_3:18;
end;
