Most of results of Bestvina and Mogilski [\textit{ Characterizing certain incomplete infinite-dimensional absolute retracts}, Michigan Math. J. \textbf{ 33} (1986), 291--313] 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 $W(X,*) = \{ (x_n)_{ n=1}^{ \infty} \in X^{ \omega}:\ x_n = * \textup{ for almost all} n\}$ is homeomorphic to a pre-Hilbert space $E$ with $E \cong \Sigma E$. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.
Categories: math.GN
