# Selectively pseudocompact groups without non-trivial convergent sequences

## TL;DR

This paper constructs a Boolean topological group in ZFC that is selectively pseudocompact but lacks non-trivial convergent sequences, addressing a major open problem in topological group theory.

## Contribution

It provides the first ZFC example of such a group with selective pseudocompactness properties and no non-trivial convergent sequences, answering longstanding questions.

## Key findings

- Constructed a Boolean topological group with selective pseudocompactness.
- Showed the group has no non-trivial convergent sequences.
- Proved free precompact Boolean groups over certain spaces contain no infinite compact subsets.

## Abstract

The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first author. As a by-product, we show that the free precompact Boolean group over any disjoint sum of maximal countable spaces contains no infinite compact subsets.