
theorem
  for T be T_0 non empty TopSpace holds card the carrier of T c= card
  the topology of T
proof
  let T be T_0 non empty TopSpace;
  defpred P[Element of T, set] means $2 = [#]T \ Cl {$1};
A1: for e be Element of T ex u be Element of the topology of T st P[e, u]
  proof
    let e be Element of T;
    reconsider u = [#]T \ Cl {e} as Element of the topology of T
           by PRE_TOPC:def 2,def 3;
    take u;
    thus thesis;
  end;
  consider f be Function of the carrier of T, the topology of T such that
A2: for e be Element of T holds P[e, f.e] from FUNCT_2:sch 3(A1);
A3: f is one-to-one
  proof
    let x1,x2 be object;
    assume that
A4: x1 in dom f & x2 in dom f and
A5: f.x1 = f.x2;
    reconsider y1 = x1, y2 = x2 as Element of T by A4;
    (Cl {y1})` = [#]T \ Cl {y1} by SUBSET_1:def 4
      .= f.x2 by A2,A5
      .= [#]T \ Cl {y2} by A2
      .= (Cl {y2})` by SUBSET_1:def 4;
    then Cl {y2} c= Cl {y1} & Cl {y1} c= Cl {y2} by SUBSET_1:12;
    hence thesis by XBOOLE_0:def 10,YELLOW_8:23;
  end;
  dom f = the carrier of T & rng f c= the topology of T by FUNCT_2:def 1;
  hence thesis by A3,CARD_1:10;
end;
