We extend the definition of Bockstein basis
σ
(
G
)
\sigma (G)
to nilpotent groups
G
G
. A metrizable space
X
X
is called a Bockstein space if
dim
G
(
X
)
=
sup
{
dim
H
(
X
)
|
H
∈
σ
(
G
)
}
\operatorname {dim}_G(X) = \sup \{\operatorname {dim}_H(X) | H\in \sigma (G)\}
for all Abelian groups
G
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
X
be a Bockstein space. If
G
G
is nilpotent, then
dim
G
(
X
)
≤
1
\operatorname {dim}_G(X) \leq 1
if and only if
sup
{
dim
H
(
X
)
|
H
∈
σ
(
G
)
}
≤
1
\sup \{\operatorname {dim}_H(X) | H\in \sigma (G)\}\leq 1
.
Theorem 0.2.
X
X
is a Bockstein space if and only if
dim
Z
(
l
)
(
X
)
=
dim
Z
^
(
l
)
(
X
)
\operatorname {dim}_{{\mathbf {Z}}_{(l)}} (X) = \operatorname {dim}_{\hat {Z}_{(l)}}(X)
for all subsets
l
l
of prime numbers.