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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...
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