reserve T, T1, T2 for TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  A1 for Subset of T1,
  A2 for Subset of T2,
  TM, TM1, TM2 for metrizable TopSpace,
  Am, Bm for Subset of TM,
  Fm, Gm for Subset-Family of TM,
  C for Cardinal,
  iC for infinite Cardinal;

theorem Th4:
  T1,T2 are_homeomorphic implies weight T1 = weight T2
proof
  assume
A1: T1,T2 are_homeomorphic;
  per cases;
  suppose
A2: [#]T1={} or [#]T2={};
A3: T1 is empty iff T2 is empty by A1,YELLOW14:18;
    then weight T1=0 by A2,Lm3;
    hence thesis by A2,A3,Lm3;
  end;
  suppose
    T1 is non empty & T2 is non empty;
    then weight T2 c=weight T1 & weight T1 c=weight T2 by A1,Lm2;
    hence thesis;
  end;
end;
