Weakly Lindelof determined Banach spaces not containing ℓ1(N)\ell^1(N)

The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing \ell^{1}(\NN ). Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing \ell^{1}(\NN ) and not being itself a W.C.G. space