We present a wide class of reflexive, precompact, non-compact, Abelian topological groupsG determined by three requirements. They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G are finite. We also show that many (non-reflexive) precompact Abelian ...

If X is a Tychonoff space, C(X) its ring of real-valued continuous functions. In this paper, we study non-essential ideals in C(X). Let a be a infinite cardinal, then X is called a-Kasch (resp. ā-Kasch) space if given any ideal (resp. z-ideal) I with gen (I) < a then I is a non-essential ideal. We show that X is an א0-Kasch space if and only if X is an almost P -space and X is an א1-Kasch space...

In this article we study the arcwise connected component in the Pontryagin dual of an abelian topological group. It is clear that the set of continuous characters that can be lifted over the reals is contained in the arcwise connected component of the dual group. We show that the converse is true if all arcs in the character group are equicontinuous sets. This property is present in Pontryagin ...

We consider when one-to-one continuous mappings can improve normalitytype and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space X there is a μ such that X can be condensed onto a normal (σ-compact) space if and only if there is no measurable cardinal. For any Tychonoff space X and any cardinal ν there is a Tychonoff space M which preserv...