Some pseudocompact-like properties in certain topological groups

TL;DR

This paper constructs specific topological groups in ZFC to explore properties related to pseudocompactness and countable compactness, demonstrating their non-equivalence and providing new examples in the field.

Contribution

It provides the first ZFC constructions of countably compact groups without non-trivial convergent sequences and of selectively pseudocompact groups not countably pracompact, clarifying their relationship.

Findings

Constructed a countably compact group without non-trivial convergent sequences in ZFC.

Created a selectively pseudocompact group that is not countably pracompact in ZFC.

Demonstrated a group with all powers selectively pseudocompact but not countably pracompact, assuming a selective ultrafilter.

Abstract

We construct in ZFC a countably compact group without non-trivial convergent sequences of size $2^{\mathfrak{c}}$, answering a question of Bellini, Rodrigues and Tomita. We also construct in ZFC a selectively pseudocompact group which is not countably pracompact, showing that these two properties are not equivalent in the class of topological groups. Using the same technique, we construct a group which has all powers selectively pseudocompact but is not countably pracompact, assuming the existence of a selective ultrafilter.