New Hindman spaces

TL;DR

This paper introduces a new combinatorial method to analyze Hindman spaces, characterizes certain ideals related to these spaces, and constructs examples under the continuum hypothesis that answer open questions in topology.

Contribution

It develops a novel approach linking topological properties of Hindman spaces to the Kat
10ov order of ideals, providing new characterizations and explicit constructions.

Findings

Characterized $F_\sigma$ ideals with Hindman spaces not being $ extit{I}$-spaces

Reduced a topological question to Kat
10ov order comparison of ideals

Constructed a Hindman space not an $ extit{I}_{1/n}$-space under continuum hypothesis

Abstract

We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Kat\v{e}tov order of ideals on $\mathbb{N}$. We also provide two applications of the method. (1) We characterize $F_\sigma$ ideals $\mathcal{I}$ for which there is a Hindman space which is not an $\mathcal{I}$-space under the continuum hypothesis. This reduces a topological question of Albin L. Jones about consistency of existence of a Hindman space which is not van der Waerden to the question whether the ideal of all non AP-sets is not below the ideal of all non IP-sets in the Kat\v{e}tov order. (2) Under the continuum hypothesis, we construct a Hindman space which is not an $\mathcal{I}_{1/n}$-space. This answers a question posed by Jana Fla\v{s}kov\'{a} at the 22nd Summer Conference on Topology and its Applications.