reserve T,T1,T2,S for non empty TopSpace;

theorem Th9:
  for B being Subset of TOP-REAL 2 st B={0.TOP-REAL 2} holds B`<>
  {} & (the carrier of TOP-REAL 2)\B<>{}
proof
  let B be Subset of TOP-REAL 2;
  assume
A1: B={0.TOP-REAL 2};
  now
    assume |[0,1]| in B;
    then |[0,1]|`2=0 by A1,Th3,TARSKI:def 1;
    hence contradiction by EUCLID:52;
  end;
  then |[0,1]| in (the carrier of TOP-REAL 2) \ B by XBOOLE_0:def 5;
  hence thesis by SUBSET_1:def 4;
end;
