On iterated inverse limits

Let a compact Hausdorff space X be the limit of a cofinite inverse system of compact Hausdorff spaces Xλ, X=limλXλ. Then it is possible to express every Xλ as the limit of an inverse system of compact polyhedra Yλμ, Xλ=limμYλμ, in such a way that the spaces Yν=Yλμ can be organized in an inverse system with limνYν=limλ limμYλμ. Using ANR-resolutions, the result is generalized to non-compact spaces.