No interesting sequential groups

TL;DR

This paper demonstrates the independence of the existence of certain sequential topological groups from ZFC, showing that no intermediate sequential groups exist in some models, and resolves related open questions.

Contribution

It constructs a model of set theory where no intermediate sequential topological groups exist, answering longstanding open questions and showing independence from ZFC.

Findings

No intermediate sequential groups exist in the constructed model

Every countably compact sequential group can be Fréchet-Urysohn in this model

The results answer multiple open questions in topology and set theory

Abstract

We prove that it is consistent with ZFC that no sequential topological groups of intermediate sequential orders exist. This shows that the answer to a 1981 question of P.~Nyikos is independent of the standard axioms of set theory. The model constructed also provides consistent answers to several questions of D.~Shakhmatov, S.~Todor\v{c}evi\'c and Uzc\'ategui. In particular, we show that it is consistent with ZFC that every countably compact sequential group is Fr\'echet-Urysohn.