Pseudocompactness and ultrafilters - Fraga, Juliane Trianon (2023)
This work presents advances obtained in the theory of topological groups with pseudocompact-like properties. We construct in ZFC a countably compact group without non-trivial convergent sequences of size 2^c. We also construct in ZFC a selectively pseudocompact group which is not countably pracompact. Using the same technique, we construct a group which has all powers selectively pseudocompact but is not countably pracompact, assuming the existence of a single selective ultrafilter. Naturally, a question similar to that asked by Comfort in 1990 for countably compact groups can also be asked for countably pracompact groups: for which cardinals \\alpha is there a topological group G such that G^{\\gamma} is countably pracompact for all cardinals \\gamma < \\alpha, but G^{\\alpha} is not countably pracompact? In this work we construct such group in the case \\alpha = \\omega, assuming the existence of c incomparable selective ultrafilters, and in the case \\alpha = \\kappa^{+}, with \\omega \\leq \\kappa \\leq 2^c, assuming the existence of 2^c incomparable selective ultrafilters. We also construct an Abelian, torsion-free, non-divisible topological group which is compact, and show that for every Abelian group G, Z \\times G does not admit a p-compact group topology for any free ultrafilter p. We show that the previous result is also true when we replace Z by a subgroup of Q that is r-divisible for every prime r except exactly one of them. Finally, we show that there exists a p-compact group topology on Q^(c) without non-trivial convergent sequences for which we find a closed subgroup H \\subset Q^(c) which contains an element not divisible (in H) by any natural.