Coarse amenability and discreteness

TL;DR

This paper explores the dualization of paracompactness into the coarse category using $R$-disjointness, introducing the concept of countable asymptotic dimension and relating it to Property A and finite asymptotic dimension.

Contribution

It generalizes finite decomposition complexities to countable asymptotic dimension and establishes its relationship with Property A and finite asymptotic dimension.

Findings

Countable asymptotic dimension is a subclass of spaces with Property A.

Provides a necessary and sufficient condition for countable asymptotic dimension to imply finite asymptotic dimension.

Unifies concepts of finite decomposition complexity and straight finite decomposition complexity.

Abstract

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of G.Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of G.Yu and straight finite decomposition complexity of Dranishnikov-Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.