In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is Kσ, the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is Kσ (since it is countable) but not compactly generated (since compact subsets are finite). Beros showe...

We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-LindenstraussWeiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.

Under the axiom of choice, every first countable space is a FréchetUrysohn space. Although, in its absence even R may fail to be a sequential space. Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces or the subspaces of R, are classes of Fréchet-Urysohn or sequential spaces. In this context, i...

An infinite countable Steiner triple system is called universal if any countable Steiner triple system can be embedded into it. The main result of this paper is the proof of non-existence of a universal Steiner triple system. The fact is proven by constructing a family S of size 2 of infinite countable Steiner triple systems so that no finite Steiner triple system can be embedded into any of th...

In this paper we prove that it consistent to have a two-point set in a model of ZF in which the real line cannot be well-ordered. We prove two results related to a construction of Chad of a two-point set inside the countable union of concentric circles. We show that if the reals are the countable union of countable sets, then every well-orderable set of reals is countable. However, it is consis...

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti’s central argument against countable additivity and provide a new Dutch book proof of the principle, to argue that if we accept the Dutch book foundations of subjective probability, countable additivity is a...

Let T be a 1-sorted structure. The functor TotFamT yielding a family of subsets of T is defined by: (Def. 1) TotFamT = 2 carrier of T . The following proposition is true (1) For every set T and for every family F of subsets of T holds F is countable iff F c is countable. Let us note that Q is countable. The scheme FraenCoun11 concerns a unary predicate P, and states that: {{n};n ranges over ele...