Strengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

We develop the basics of an analogue of descriptive set theory for functions on a Polish space X. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

We investigate the continuous model theory of ultrametric spaces of diameter ≤ 1. There is no universal Polish ultrametric space of diameter 1; but there is a Polish ultrametric space, Umax, taking distances in Q∩[0, 1], which is universal for all such Polish ultrametric spaces. We show that in the continuous theory of Umax, nonforking is characterized by a stable independence relation, which i...

The study of simply definable equivalence relations, and in particular of Borei and analytic ones, on Polish spaces, has attracted a lot of attention in descriptive set theory since 1970, when the first fundamental result in the subject, Silver's theorem, was proved. One motivation has certainly been the conjecture of Vaught in model theory, one of the oldest still open problems in mathematical...

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum functi...

For Ω a Polish space we consider operator semigroups on the space Cb(Ω) of bounded, continuous functions and on the space M (Ω) of bounded Borel measures. The important fact here is that the dual of Cb(Ω) with the strict topology is M (Ω). We pursue the approach via bi-continuous semigroups, but we also show that the class of bi-continuous semigroups on Cb(Ω) with respect to the compact-open to...