The pseudocompact extension $\alpha X$

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...

Maximal pseudocompact spaces

Maximal pseudocompact spaces (i.e. pseudocompact spaces possessing no strictly stronger pseudocompact topology) are characterized. It is shown that submaximal pseudocompact spaces whose pseudocompact subspaces are closed need not be maximal pseudocompact. Various techniques for constructing maximal pseudocompact spaces are described. Maximal pseudocompactness is compared to maximal feeble compa...

Extremal α-pseudocompact abelian groups

Let α be an infinite cardinal. Generalizing a recent result of Comfort and van Mill, we prove that every α-pseudocompact abelian group of weight > α has some proper dense α-pseudocompact subgroup and admits some strictly finer α-pseudocompact group topology. AMS classification numbers: Primary 22B05, 22C05, 40A05; Secondary 43A70, 54A20.