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Topological AE(0)-groups

Alex Chigogidze (2001)

Fundamenta Mathematicae

We investigate topological AE(0)-groups, a class which contains the class of Polish groups as well as the class of all locally compact groups. We establish the existence of a universal AE(0)-group of a given weight as well as the existence of a universal action of an AE(0)-group of a given weight on an AE(0)-space of the same weight. A complete characterization of closed subgroups of powers of the symmetric group S ∞ is obtained. It is also shown that every AE(0)-group is Baire isomorphic to a product...

Topological groups with Rokhlin properties

Eli Glasner, Benjamin Weiss (2008)

Colloquium Mathematicae

In his classical paper [Ann. of Math. 45 (1944)] P. R. Halmos shows that weak mixing is generic in the measure preserving transformations. Later, in his book, Lectures on Ergodic Theory, he gave a more streamlined proof of this fact based on a fundamental lemma due to V. A. Rokhlin. For this reason the name of Rokhlin has been attached to a variety of results, old and new, relating to the density of conjugacy classes in topological groups. In this paper we will survey some of the new developments...

Topological type of weakly closed subgroups in Banach spaces

Tadeusz Dobrowolski, Janusz Grabowski, Kazuhiro Kawamura (1996)

Studia Mathematica

The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in ℓ 1 which are interesting from the Banach space theory point of view are discussed.

Topology of the isometry group of the Urysohn space

Julien Melleray (2010)

Fundamenta Mathematicae

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space ℓ²(ℕ). The proof is based on a lemma about extensions of metric spaces by finite metric spaces, which we also use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise, the group...

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