On the doubling condition in the infinite-dimensional setting

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.