2,794 research outputs found
Grey subsets of Polish spaces
We develop the basics of an analogue of descriptive set theory for functions
on a Polish space . 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
Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces
Assuming the existence of certain large cardinal numbers, we prove that for
every projective filter over the set of natural numbers,
-bases in Banach spaces have continuous coordinate functionals. In
particular, this applies to the filter of statistical convergence, thereby we
solve a problem by V. Kadets (at least under the presence of certain large
cardinals). In this setting, we recover also a result of Kochanek who proved
continuity of coordinate functionals for countably generated filters (Studia
Math., 2012).Comment: 10 p
Metrizable universal minimal flows of Polish groups have a comeagre orbit
We prove that, whenever is a Polish group with metrizable universal
minimal flow , there exists a comeagre orbit in . It then follows
that there exists an extremely amenable, closed, coprecompact of such
that
On Roeckle-precompact Polish group which cannot act transitively on a complete metric space
We study when a continuous isometric action of a Polish group on a complete
metric space is, or can be, transitive. Our main results consist of showing
that certain Polish groups, namely and
, such an action can never be transitive (unless the
space acted upon is a singleton). We also point out "circumstantial evidence"
that this pathology could be related to that of Polish groups which are not
closed permutation groups and yet have discrete uniform distance, and give a
general characterisation of continuous isometric action of a Roeckle-precompact
Polish group on a complete metric space is transitive. It follows that the
morphism from a Roeckle-precompact Polish group to its Bohr compactification is
surjective
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