Abstract
Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover \( \mathcal{U} \) of X there is a sequence of maps (f n : X → X) nεgw such that each f n is \( \mathcal{U} \)-near to the identity map of X and the family {f n (X)} n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < \( \mathfrak{d} \) is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z ∞-set in X.
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Banakh, T., Zarichnyy, I. Topological groups and convex sets homeomorphic to non-separable Hilbert spaces. centr.eur.j.math. 6, 77–86 (2008). https://doi.org/10.2478/s11533-008-0005-0
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DOI: https://doi.org/10.2478/s11533-008-0005-0