We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base a...

Let G be a closed subgroup of S∞ and X be a Polish G-space. To each x ∈ X we associate an admissible set Ax and show how questions about X which involve Baire category, can be formalized in Ax.

This paper extends the theory of turbulence of Hjorth to certain classes of equivalence relations that cannot be induced by Polish actions. It applies this theory to analyze the quasi-isometry relation and finite Gromov-Hausdorff distance relation in the space of isometry classes of pointed proper metric spaces, called the Gromov space.

We prove that for each Polish space X , the space C(X) of continuous real-valued functions on X satisfies (a strong version of) the Pytkeev property, if endowed with the compactopen topology. We also consider the Pytkeev property in the case where C(X) is endowed with the topology of pointwise convergence.

Simple argument in favour of unitarity, to all orders, of space-like noncommutative field theory is given. ∗supported by the grant no. 1 P03B 021 28 of Polish Minister of Science

In this paper we establish under suitable conditions the existence of measurable selectors for Gs-valued multifunctions. In particular we prove that a measurable partition of a Polish space into G6 sets admits a Borel selector.

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.