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

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-dimensional 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 .: x n = * for almost all n} is homeomorphic to a pre-Hilbert space E with E . .E. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.