  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.
