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

Most of results of Bestvina and Mogilski [\textit{Characterizing certain incomplete infinite-dimensional absolute retracts}, Michigan Math. J. $\mathbf{\text{33}}$ (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\mathrm{‘}$w${}^{\mathrm{\prime }}isthetopologicalweight)foreachopennonemptysubset$U$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 $E\cong \mathrm{\Sigma }E$. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.