In this note we show that the p-rank of Ext(A,B) for countable torsion-free abelian groups A and B is either countable or the size of the continuum.

In this paper we introduce a slightly stronger form of countable dense homogeneity that for Polish spaces can be characterized topologically in a natural way. Along the way, we generalize theorems obtained by Bennett and Ungar on countable dense homogeneity.

Improving a result from the paper “Spaces in countable web” by Yoshikazu Yasui and Zhi-min Gao we show that Tychonoff spaces in countable discrete web may contain closed discrete subsets of arbitrarily big cardinality.

We study the possible cardinalities of the divisible part of G/K when the cardinality of K is known and when, for all countable subgroups C of G, the divisible part of G/C is countable.

Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X, S, μ) and A a locally compact second countable abelian group. A cocycle c : G × X 7−→ A for T disperses if limg→∞ c(g, ·) − α(g) = ∞ in measure for every map α : G −→ A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A0 ⊂ A such that the comp...