On the doubling condition in the infinite-dimensional setting

TL;DR

This paper investigates whether infinite-dimensional topological spaces can be made homogeneous in the Coifman--Weiss sense, concluding negatively, with the infinite torus as a key example.

Contribution

It provides a systematic approach to the homogeneity problem in infinite-dimensional spaces, confirming the negative answer and highlighting the infinite torus as a primary example.

Findings

Infinite-dimensional spaces cannot be made homogeneous in the Coifman--Weiss sense

The infinite torus $ ext{T}^ ext{ω}$ exemplifies the space with this property

The approach clarifies limitations of homogeneity in infinite-dimensional topology

Abstract

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman--Weiss sense. The answer to the examined question is negative, as expected. Our leading representative of spaces with this property is $\mathbb{T}^{\omega} = \mathbb{T} \times \mathbb{T} \times \cdots$ with the natural product topology.