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

Open Mathematics (2008)

- Volume: 6, Issue: 1, page 77-86
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topTaras Banakh, and Igor Zarichnyy. "Topological groups and convex sets homeomorphic to non-separable Hilbert spaces." Open Mathematics 6.1 (2008): 77-86. <http://eudml.org/doc/269125>.

@article{TarasBanakh2008,

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.},

author = {Taras Banakh, Igor Zarichnyy},

journal = {Open Mathematics},

keywords = {Hilbert manifold; convex set; topological group; Z∞-set},

language = {eng},

number = {1},

pages = {77-86},

title = {Topological groups and convex sets homeomorphic to non-separable Hilbert spaces},

url = {http://eudml.org/doc/269125},

volume = {6},

year = {2008},

}

TY - JOUR

AU - Taras Banakh

AU - Igor Zarichnyy

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

JO - Open Mathematics

PY - 2008

VL - 6

IS - 1

SP - 77

EP - 86

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/269125

ER -

## References

top- [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 Zbl0967.57021
- [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 Zbl0472.57009
- [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 Zbl0468.57015
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.