Set-Theoretical Problems in Asymptology

TL;DR

This paper compiles open set-theoretic problems in large-scale topology, focusing on coarse structures, ultrafilters, and properties of coarse spaces, highlighting key questions and challenges in asymptology.

Contribution

It introduces and discusses several open problems in asymptology related to coarse structures, ultrafilters, and coarse groups, advancing the understanding of large-scale topology.

Findings

Identification of critical cardinalities in coarse structures on ω

Analysis of properties of the Higson corona in coarse spaces

Exploration of extremal properties of coarse groups

Abstract

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $\omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $\omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.