Convergence in topological groups and the Cohen reals

TL;DR

This paper demonstrates the consistency of having an uncountable sequential topological group with intermediate sequential order, while no such countable groups exist, using Cohen reals and diamond principles.

Contribution

It introduces a new consistency result linking Cohen reals, diamond principles, and the structure of sequential groups in topology.

Findings

Uncountable sequential groups of intermediate order exist under certain set-theoretic assumptions.

No countable sequential groups of intermediate order exist in the same model.

Adding  Cohen reals affects the existence of such groups.

Abstract

We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding $\omega_2$ Cohen reals to a model of $\diamondsuit$.