Central European Journal of Mathematics

, Volume 6, Issue 1, pp 77–86 | Cite as

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

Research Article

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 (fn: XX)nεgw such that each fn is \( \mathcal{U} \)-near to the identity map of X and the family {fn(X)}nω 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 AX of density dens(A) < dens(X) is a Z-set in X.

Keywords

Hilbert manifold convex set topological group Z-set 

MSC

57N17 57N20 

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Copyright information

© © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Instytut MatematykiAkademia ŚwiętokrzyskaKielcePoland
  2. 2.Department of MathematicsIvan Franko National University of LvivLvivUkraine

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