Unbounded towers and products

TL;DR

This paper explores the existence and properties of certain sets of reals with combinatorial covering properties, establishing new results under the assumption of an unbounded tower without extra set-theoretic assumptions.

Contribution

It proves the existence of nontrivial sets satisfying $ ext{S}_1( ext{Gamma}, ext{Gamma})$ in all finite powers assuming an unbounded tower, and characterizes when products of sets satisfy $ ext{Omega}inom{ ext{Gamma}}{ ext{Gamma}}$.

Findings

Existence of nontrivial sets satisfying $ ext{S}_1( ext{Gamma}, ext{Gamma})$ under unbounded tower assumption.

Finite products of certain sets satisfy $ ext{Omega}inom{ ext{Gamma}}{ ext{Gamma}}$.

Necessary and sufficient conditions for productively $ ext{Omega}inom{ ext{Gamma}}{ ext{Gamma}}$ sets.

Abstract

We investigate products of sets of reals with combinatorial covering properties. A topological space satisfies $\mathsf{S}_1(\Gamma,\Gamma)$ if for each sequence of point-cofinite open covers of the space, one can pick one element from each cover and obtain a point-cofinite cover of the space. We prove that, if there is an unbounded tower, then there is a nontrivial set of reals satisfying $\mathsf{S}_1(\Gamma,\Gamma)$ in all finite powers. In contrast to earlier results, our proof does not require any additional set-theoretic assumptions. A topological space satisfies $\Omega\choose\Gamma$ (also known as Gerlits--Nagy's property $\gamma$) if every open cover of the space such that each finite subset of the space is contained in a member of the cover, contains a point-cofinite cover of the space. We investigate products of sets satisfying $\Omega\choose\Gamma$ and their relations to other classic combinatorial covering properties. We show that finite products of sets with a certain combinatorial structure satisfy $\Omega\choose\Gamma$ and give necessary and sufficient conditions when these sets are productively $\Omega\choose\Gamma$.