Bockstein theorem for nilpotent groups

We extend the definition of Bockstein basis $\sigma(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a Bockstein space if $\operatorname{dim}_G(X) = \sup\{\operatorname{dim}_H(X) \vert H\in \sigma(G)\}$ for all Abelian groups $G$. The Bockstein First Theorem says that all compact spaces are Bockstein spaces.

Here are the main results of the paper:

Theorem 0.1. Let $X$ be a Bockstein space. If $G$ is nilpotent, then $\operatorname{dim}_G(X) \leq 1$ if and only if $\sup\{\operatorname{dim}_H(X) \vert H\in\sigma(G)\}\leq 1$.

Theorem 0.2. $X$ is a Bockstein space if and only if $\operatorname{dim}_{{\mathbf{Z}}_{(l)}} (X) = \operatorname{dim}_{\hat{Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers.