Intermediate-value property for Assouad dimension by its inversion invariance

Jouni Luukkainen

doi:10.1016/j.topol.2021.107700

Abstract

We prove that if X is a metric space of Assouad dimension s∈(0,∞), then for every α∈[0,s] there is a countable subset of X of Assouad dimension α. We reduce this to the special case of bounded X due to Wang and Wen in 2016 by using the inversion invariance of Assouad dimension.

Highlights

Let dimA X denote the Assouad dimension of a metric space (X, d) in the sense of [3]
Otherwise even dimA X = 0 for the bounded but not totally bounded space X in [4, Remark 1].) We observe that here the requirement of countability is redundant as every metric space of finite Assouad dimension is separable and as the closure operator preserves the Assouad dimension of a subset by [3, A.5(2)]
It is desirable to get the result of Wang and Wen for an arbitrary, possibly unbounded metric space X and to have just such a proof for the general case which is a reduction to the special case of bounded X

Summary

Introduction

Let dimA X denote the Assouad dimension of a metric space (X, d) in the sense of [3]. Wang and Wen [4] proved that if X is a bounded metric space with s = dimA X ∈ (0, ∞), for every α ∈ (0, s) there is a countable subset F of X with dimA F = α. Otherwise even dimA X = 0 for the bounded but not totally bounded space X in [4, Remark 1].) We observe that here the requirement of countability is redundant as every metric space of finite Assouad dimension is separable and as the closure operator preserves the Assouad dimension of a subset by [3, A.5(2)].

Results

Conclusion