On asymptotic dimension and a property of Nagata

TL;DR

The paper proves that metric spaces with finite asymptotic dimension are coarsely equivalent to spaces satisfying a Nagata property, resolving a known open problem in topology.

Contribution

It establishes a coarse equivalence between spaces of bounded asymptotic dimension and spaces with a specific Nagata property, solving an open problem.

Findings

Every metric space with asymptotic dimension at most n is coarsely equivalent to a Nagata space.

The Nagata property involves a specific relation among points and distances in the space.

This result confirms a conjecture posed in the literature.

Abstract

In this note we prove that every metric space $(X, d)$ of asymptotic dimmension at most $n$ is coarsely equivalent to a metric space $(Y, D)$ that satisfies the following property of Nagata: For every $n+2$ points $y_1,..., y_{n+2}$ in $Y$ and for every $x$ in $Y$ there exist two different $i,j$ such that $D(y_i,y_j)\le D(x,y_i)$. This solves problem 1400 of the book Open problems in Topology II.