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

TL;DR

This paper extends results on strong Z-sets and absorbing sets from separable to nonseparable ANRs, characterizing when such spaces embed in Hilbert manifolds and describing their intrinsic structure.

Contribution

It generalizes key theorems on ANRs and absorbing sets to nonseparable cases and provides an intrinsic characterization of manifolds modeled on specific pre-Hilbert spaces.

Findings

ANRs with certain properties embed in Hilbert manifolds

Weak products of ARs are homeomorphic to pre-Hilbert spaces

Intrinsic characterization of manifolds modeled on these spaces

Abstract

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.