Universal spaces of non-separable absolute Borel classes

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

“…So, in a sense, the present paper resolves an old problem posed in the known list of problems [13]. Partial results in this direction were also obtained in 2003 by Sakai and Yaguchi [11] and by Mine [14] in 2006.…”

“…C-absorbing AR's for C = M α (m) as well as for C = A α (m) (with arbitrary α and m) were constructed by Mine [14] in 2006. For ordinals α greater than 1 (and uncountable cardinals m) and C ∈ {M f α (m), A f α (m)} there are C-absorbing sets if only there are spaces X satisfying (U1) and (U2) (thanks to Proposition 7.9).…”

Characterizations of manifolds modeled on absorbing sets in non-separable Hilbert spaces and the discrete cells property

“…We will study these properties in Section 4. We say that a class C is topological if every space homeomorphic to some member of C also belongs to C, and C is closed hereditary if any closed subspace of some member of C also belongs to C. In Section 5, using the discrete cells property, we shall characterize infinite-dimensional manifolds modeled on absorbing sets in Hilbert spaces as follows, which generalizes and improves the previous results in [4,13,8]: Theorem 1.1. Let C be a topological and closed hereditary class, and Ω be a C-absorbing set in 2 (κ).…”

“…Moreover, M 0 is the class of compact spaces, M 1 is the one of completely metrizable spaces, and A 1 is the class of σ-locally compact spaces. It is known that for α ≥ 1, there exist absorbing sets for the absolute Borel classes A α (κ) and M α (κ) in the Hilbert space 2 (κ) [4,13,8]. The spaces 2 (κ) × f 2 and f 2 (κ) × I ω , where I ω is the Hilbert cube, can be regarded as absorbing sets for M 1 (κ) and A 1 (κ) in 2 (κ) respectively.…”

“…For simplicity, let B α denote one of the absolute Borel classes A α or M α , and let Ω α (κ) be a B α (κ)-absorbing set in 2 (κ). For α ≥ 2, M. Bestvina, J. Mogilski, and K. Mine [4,8] gave characterizations for manifolds modeled on Ω α (κ):…”