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 Th20:
  weight TM c= iC iff for Am st Am is discrete holds card Am c= iC
proof
  hereby
    assume weight TM c=iC;
    then
    for A be Subset of TM st A is closed & A is discrete holds card A c=iC
    by Th19;
    hence for A be Subset of TM st A is discrete holds card A c=iC by Th14;
  end;
  assume for A be Subset of TM st A is discrete holds card A c=iC;
  then for F be Subset-Family of TM st F is open & not{} in F & for A,B be
  Subset of TM st A in F & B in F & A<>B holds A misses B holds card F c=iC by
Th15;
  then density TM c=iC by Lm6;
  hence thesis by Lm7;
end;
