
theorem Th3:
  for X be set, A be Subset-Family of X holds TopStruct (#X,UniCl
    FinMeetCl A#) is TopSpace-like
proof
  let X be set;
  let A be Subset-Family of X;
  per cases;
  suppose
A1: X = {};
    set T = TopStruct (#X, UniCl FinMeetCl A#);
    the carrier of T in FinMeetCl A & FinMeetCl A c= UniCl FinMeetCl A by
CANTOR_1:1,8;
    hence the carrier of T in the topology of T;
    hence for a being Subset-Family of T st a c= the topology of T holds union
    a in the topology of T by A1;
    thus thesis by A1;
  end;
  suppose
    X <> {};
    hence thesis by CANTOR_1:15;
  end;
end;
