Topological groups and convex sets homeomorphic to non-separable Hilbert spaces |
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Authors: | Taras Banakh Igor Zarichnyy |
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Institution: | (1) Instytut Matematyki, Akademia Świętokrzyska, Kielce, Poland;(2) Department of Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine |
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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 of X there is a sequence of maps (f
n
: X → X)
nεgw
such that each f
n
is -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) < 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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Keywords: | Hilbert manifold convex set topological group Z∞ -set |
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