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 Th21:
  weight TM c= iC iff 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
proof
  hereby
    assume weight TM c=iC;
    then for A be Subset of TM st A is discrete holds card A c=iC by Th20;
    hence 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;
  end;
  assume 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;
  then density TM c=iC by Lm6;
  hence thesis by Lm7;
end;
