
theorem
  for T being non empty TopSpace st T is sober locally-compact holds T is Baire
proof
  let T be non empty TopSpace such that
A1: T is sober locally-compact;
  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 open dense;
A4: F is open
  by A3;
  for X being Subset of T st X in F holds X is dense by A3;
  then
A5: F is open dense by A4;
  reconsider I = Intersect F as Subset of T;
  take I;
  thus I = Intersect F;
  per cases;
  suppose
A6: F <> {};
    for Q being Subset of T st Q <> {} & Q is open holds (Intersect F) meets Q
    proof
      let Q be Subset of T;
      assume that
A7:   Q <> {} and
A8:   Q is open;
      consider S being irreducible Subset of T such that
A9:   for V being Subset of T st V in F holds S /\ Q meets V by A1,A2,A5,A6,A7
,A8,Th43;
      consider p being Point of T such that
A10:  p is_dense_point_of S and
      for q being Point of T st q is_dense_point_of S holds p = q by A1;
      S is closed by YELLOW_8:def 3;
      then
A11:  S = Cl{p} by A10,YELLOW_8:16;
A12:  for Y being set holds Y in F implies p in Y & p in Q
      proof
        let Y be set;
        assume
A13:    Y in F;
        then reconsider Y1 = Y as Subset of T;
        S /\ Q meets Y1 by A9,A13;
        then
A14:    S /\ Q /\ Y1 <> {};
        now
          assume not p in Q /\ Y1;
          then p in (Q /\ Y1)` by XBOOLE_0:def 5;
          then {p} c= (Q /\ Y1)` by ZFMISC_1:31;
          then
A15:      Cl{p} c= Cl(Q /\ Y1)` by PRE_TOPC:19;
          Y1 is open by A3,A13;
          then Q /\ Y1 is open by A8;
          then (Q /\ Y1)` is closed;
          then S c= (Q /\ Y1)` by A11,A15,PRE_TOPC:22;
          then S misses (Q /\ Y1) by SUBSET_1:23;
          then S /\ (Q /\ Y1) = {};
          hence contradiction by A14,XBOOLE_1:16;
        end;
        hence thesis by XBOOLE_0:def 4;
      end;
      then for Y being set holds Y in F implies p in Y;
      then
A16:  p in Intersect F by SETFAM_1:43;
      ex Y being object st Y in F by A6,XBOOLE_0:def 1;
      then p in Q by A12;
      then (Intersect F) /\ Q <> {} by A16,XBOOLE_0:def 4;
      hence thesis;
    end;
    hence thesis by TOPS_1:45;
  end;
  suppose
    F = {};
    then Intersect F = [#]T by SETFAM_1:def 9;
    hence thesis;
  end;
end;
