Normal systems over ANRs, rigid embeddings and nonseparable absorbing sets

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.

💡 Research Summary

The paper by Piotr Niemiec extends the seminal work of Bestvina and Mogilski on strong Z‑sets and absorbing sets from the separable (countable) setting to the non‑separable (uncountable) realm. The author introduces the notion of a “normal system” – a collection of subsets satisfying three closure properties (M1–M3) that generalize both the definition of strong Z‑sets and the strong discrete approximation property (SDAP). By interpreting these collections as Michael families, the author proves that normal systems enjoy a small‑maps approximation property (SMAP): there exists an open cover U of the target space Y such that the set of U‑small maps S_Y(X,U) lies in the closure of a given family D⊂C(X,Y). This SMAP condition is shown to be equivalent to the density of D in the function space, and it is hereditary for closed subsets.

Using normal systems and SMAP, the paper proves a key embedding theorem: if an ANR X can be locally homotopy‑dense embedded in infinite‑dimensional Hilbert manifolds and every non‑empty open subset U⊂X has the same topological weight w(U)=w(X), then X itself admits a homotopy‑dense embedding into a Hilbert manifold. This result removes the separability hypothesis that was essential in earlier works and shows that the weight‑uniformity condition suffices for a global Hilbert‑manifold embedding.

The second major contribution concerns the weak product of an absolute retract (AR). For an AR X with a distinguished base point *, the weak product \