New Hindman spaces

Abstract

We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Katětov 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 Katětov 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 Flašková at the 22nd Summer Conference on Topology and its Applications.