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

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.