Invariant Ideal Axiom

TL;DR

The paper introduces the Invariant Ideal Axiom, a new set theoretic axiom that allows for a complete topological classification of countable sequential groups and characterizes their finite products.

Contribution

It presents the consistency proof of the Invariant Ideal Axiom and demonstrates its applications in classifying countable sequential groups and analyzing their finite products.

Findings

Provides a full topological classification of countable sequential groups

Characterizes the behavior of finite products of these groups

Constructs examples demonstrating the optimality of the conditions in IIA

Abstract

We introduce and prove the consistency of a new set theoretic axiom we call the \emph{Invariant Ideal Axiom}. The axiom enables us to provide (consistently) a full topological classification of countable sequential groups, as well as fully characterize the behavior of their finite products. We also construct examples that demonstrate the optimality of the conditions in \IIA, and list a number of open questions.