Abstract

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 X_i having the property that X_iτ|K| for each i in 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 X_i where dim_GX_i ≤ n for each i in N, then dim_GX ≤ n.

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