Functions with pseudocompact support

Purity of the Ideal of Continuous Functions with Pseudocompact Support

Let CΨ (X) be the ideal of functions with pseudocompact support and let kX be the set of all points in υX having compact neighborhoods. We show that CΨ (X) is pure if and only if βX−kX is a round subset of βX, CΨ (X) is a projective C(X)-module if and only if CΨ (X) is pure and kX is paracompact. We also show that if CΨ (X) is pure, then for each f ∈ CΨ (X) the ideal (f ) is a projective (flat)...

Some Properties of the Ideal of Continuous Functions with Pseudocompact Support

Let C(X) be the ring of all continuous real-valued functions defined on a completely regular T1-space. Let CΨ (X) and CK(X) be the ideal of functions with pseudocompact support and compact support, respectively. Further equivalent conditions are given to characterize when an ideal of C(X) is a P -ideal, a concept which was originally defined and characterized by Rudd (1975). We used this new ch...

Pseudocompact Spaces

Pseudocompact Totally Dense Subgroups

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K admits a proper totally dense pseudocompact subgroup, then K cannot have a torsion closed Gδ-subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H of K that contains an ω-bounded dense sub...

Non-Abelian Pseudocompact Groups

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K|-many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22 |K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudoco...