Asymptotic structures of cardinals

TL;DR

This paper explores the asymptotic structures of cardinals through balleans, characterizing their properties and invariants, and discussing ultrafilters on these structures.

Contribution

It introduces a natural order-based construction of balleans on cardinals and provides their classification, metrizability criteria, and cardinal invariants.

Findings

Characterization of balleans up to coarse equivalence

Criteria for metrizability and cellularity

Calculation of basic cardinal invariants

Abstract

A ballean is a set $X$ endowed with some family $\F$ of its subsets, called the balls, in such a way that $(X,\F)$ can be considered as an asymptotic counterpart of a uniform topological space. Given a cardinal $\kappa$, we define $\F$ using a natural order structure on $\kappa$. We characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. We conclude the paper with discussion of some special ultrafilters on cardinal balleans.