Precompact Fréchet topologies on Abelian groups

AbstractWe study precompact Fréchet topologies on countable Abelian groups. For every countable Abelian group G we introduce the notion of a γG-set and show that there is a precompact Fréchet non-metrizable topology on G if and only if there is an uncountable γG-set that separates points of G. We show that, assuming the existence of an uncountable γ-set, there is a non-metrizable precompact Fréchet topology on every countable Abelian group, and assuming p>ω1, there is a non-metrizable Fréchet topology on every countable group which admits a non-discrete topology at all. We further study the notion of a γG-set and show that the minimal size of a subset of the dual group G⁎ which is not a γG-set is the pseudointersection number p for any countable Abelian group G

On a class of pseudocompact spaces derived from ring epimorphisms

AbstractA Tychonoff space X is RG if the embedding of C(X)→C(Xδ) is an epimorphism of rings. Compact RG-spaces are known and easily described. We study the pseudocompact RG-spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The main theorems shows, how to construct a suitable maximal almost disjoint family, and apply it to obtain examples of RG-spaces that are almost compact, locally compact, non-compact, almost-P, and of Cantor Bendixon degree 2. More complicated examples of pseudocompact non-compact RG-spaces ensue

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC, 2c non-metrizable infinite pairwise nonhomeomorphic minimal sets on compact connected linearly ordered space