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Limit theorem for inverse sequences of metric spaces in extension theory

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.