We prove a limit theorem for
extension theory for metric spaces. This theorem can be put in the following way.
Suppose that K is a simplicial complex, |K| is given the weak topology, and a
metrizable space X is the limit of an inverse sequence of metrizable spaces Xi having
the property that Xiτ|K| for each i ∈ ℕ. Then Xτ|K|. This latter property means
that for each closed subset A of X and map f : A →|K|, there exists a map
F : X →|K| which is an extension of f.
As a corollary to this we get the result of Nagami that the limit of an inverse
sequence of metrizable spaces each having dimension ≤ n has dimension ≤ n. But we
get much more, as this result extends to cohomological dimension modulo an
abelian group. Hence, if G is an abelian group and X is the limit of an inverse
sequence of metrizable spaces Xi where dimGXi≤ n for each i ∈ ℕ, then
dimGX ≤ n.
