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

Open Mathematics (2008)

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

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## Abstract

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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 $𝒰$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $𝒰$ -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) < $𝔡$ 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.

## How to cite

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Taras 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

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1. [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. [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. [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. [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. [5] Toruńczyk H., Characterizing Hilbert space topology, Fund. Math., 1981, 111, 247–262 Zbl0468.57015
6. [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

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