reserve X for non empty set;

theorem
  for ET being FMT_TopSpace, B being Basis of ET holds
  Family_open_set(ET) = UniCl B
  proof
    let X be FMT_TopSpace, B be Basis of X;
    thus Family_open_set(X) c= UniCl B by Def8;
    hereby
      let t be object;
      assume t in UniCl B;
      then consider Y be Subset-Family of X such that
A1:   Y c= B & t=union Y by CANTOR_1:def 1;
      B is open;
      then B c= Family_open_set(X);
      then Y c= Family_open_set(X) by A1;
      hence t in Family_open_set(X) by A1,Th9;
    end;
  end;
