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 Th19:
  weight TM c=iC iff for Am st Am is closed & Am is discrete holds
  card Am c= iC
proof
  hereby
    assume weight TM c=iC;
    then for Fm st Fm is open & Fm is Cover of TM ex Gm st Gm c=Fm & Gm is
    Cover of TM & card Gm c=iC by Th18;
    hence for Am st Am is closed & Am is discrete holds card Am c=iC by Th13;
  end;
  assume for Am st Am is closed & Am is discrete holds card Am c=iC;
  then for Am st Am is discrete holds card Am c=iC by Th14;
  then
  for Fm st Fm is open & not{} in Fm & for Am,Bm st Am in Fm & Bm in Fm &
  Am<>Bm holds Am misses Bm holds card Fm c=iC by Th15;
  then density TM c=iC by Lm6;
  hence thesis by Lm7;
end;
