Normal systems over ANR’s, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di-mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U ) = w(X ) (where "w"is the topological weight) for each open nonempty subset U of X , then X itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W (X, ) = {(x n ) ∞ n=1 ∈ X ω : x n = for almost all n} is homeomorphic to a pre-Hilbert space E with E ～ = ΣE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.