Products of finite groups and nonmeasurable subgroups

It is proven that if $G$ is a finite group, then $G^\omega$ has $2^{\mathfrak c}$ dense nonmeasurable subgroups. Also, other examples of compact groups with dense nonmeasurable subgroups are presented.Comment: 5 page

When a totally bounded group topology is the Bohr Topology of a LCA group

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu]

Contributions to the Bohr topology by W.W. Comfort

The important rôle that W. W. Comfort played in the study of the Bohr topology is described

Locally pseudocompact topological groups

AbstractA topological group is said to be locally pseudocompact if the identity has a pseudocompact neighborhood (equivalently: if the identity has a local basis of pseudocompact neighborhoods). Such groups are locally bounded in the sense of A. Weil, so each such group G is densely embedded in an essentially unique locally compact group G (called its Weil completion). The authors present necessary and sufficient conditions of local and global nature for a locally bounded group to be locally pseudocompact, as follows. Theorem. If G is a locally bounded group with Weil completionG, then the following conditions are equivalent: 1.(i) Gis locally pseudocompact;2.(ii) G is C∗-embedded in G (i.e., βG = βG);3.(iii) G is C-embedded in G (i.e., υG = υG);4.(iv) G is M-embedded in G (i.e., γG = G);5.(v) some nonempty open subsetU of G satisfies β(clGU) = clGU;6.(vi) every bounded open subset U of G satisfies β(clGU) = clGU

Weakly pseudocompact subsets of nuclear groups

Let G be an Abelian topological group and G(+) the group G endowed with the weak topology induced by continuous characters. We say that G respects compactness (pseudocompactness, countable compactness, functional boundedness) if G and G+ have the same compact (pseudocompact, countably compact, functionally bounded) sets. The well-known theorem of Glicksberg that LCA groups respect compactness was extended by Trigos-Arrieta to pseudocompactness and functional boundedness. In this paper we generalize these results to arbitrary nuclear groups, a class of Abelian topological groups which contains LCA groups and nuclear locally convex spaces and is closed with respect to subgroups, separated quotients and arbitrary products