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 $W(X,\ast )=\{(x_n{)}_{n=1}^{\infty }\in X^{\omega }:x_n=\ast \text{for almost all}n\}$ 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.

arXiv comment26 pages