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

• Autorzy: Taras Banakh , Igor Zarichnyy
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Hilbert manifold; convex set; topological group; Z∞-set

Kategorie tematyczne

• 57N17: Topology of topological vector spaces
• 57N20: Topology of infinite-dimensional manifolds

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2008
• Tom: 6
• Numer: 1
• Strony: 77-86

Daty

wydano

2008-03-01

online

2008-02-26

Twórcy

autor

Taras Banakh

autor

Igor Zarichnyy

• Ivan Franko National University of Lviv

Bibliografia

• [1] Banakh T., Characterization of spaces admitting a homotopy dense embedding into a Hilbert manifold, Topology Appl., 1998, 86, 123–131 http://dx.doi.org/10.1016/S0166-8641(97)00117-X
• [2] Banakh T., Sakai K., Yaguchi M., Zarichnyi I., Recognizing the topology of the space of closed convex subsets of a Banach space, preprint
• [3] Dobrowolski T., Toruńczyk H., Separable complete ANR’s admitting a group structure are Hilbert manifolds, Topology Appl., 1981, 12, 229–235 http://dx.doi.org/10.1016/0166-8641(81)90001-8
• [4] van Douwen E.K., The integers and Topology, In: Kunen K., Vaughan J.E. (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 111–167
• [5] Toruńczyk H., Characterizing Hilbert space topology, Fund. Math., 1981, 111, 247–262
• [6] Vaughan J.E., Small uncountable cardinals and topology, In: van Mill J., Reed C.M. (Eds.), Open Problems in Topology, North-Holland, Amsterdam, 1990, 195–218

Identyfikatory

DOI

10.2478/s11533-008-0005-0