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

  • Taras Banakh
  • Igor Zarichnyy
Research Article


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 : XX) nεgw such that each f n is \( \mathcal{U} \)-near to the identity map of X and the family {f n (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.


Hilbert manifold convex set topological group Z-set 


57N17 57N20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Banakh T., Characterization of spaces admitting a homotopy dense embedding into a Hilbert manifold, Topology Appl., 1998, 86, 123–131zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Banakh T., Sakai K., Yaguchi M., Zarichnyi I., Recognizing the topology of the space of closed convex subsets of a Banach space, preprintGoogle Scholar
  3. [3]
    Dobrowolski T., Toruńczyk H., Separable complete ANR’s admitting a group structure are Hilbert manifolds, Topology Appl., 1981, 12, 229–235zbMATHCrossRefMathSciNetGoogle Scholar
  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, 111–167Google Scholar
  5. [5]
    Toruńczyk H., Characterizing Hilbert space topology, Fund. Math., 1981, 111, 247–262MathSciNetGoogle Scholar
  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, 195–218Google Scholar

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

Personalised recommendations