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Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

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Banakh T., Zarichnyy I.

Open Mathematics

2008

tom 6

nr 1

77-86

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.

Ograniczanie wyników

1 Open Mathematics

1 Banakh T.

1 Zarichnyy I.

1 2008

Wyniki wyszukiwania

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces