On Inverse Limits of Compact Spaces. Correction of a Proof

For a compact Hausdorff space X and an ANR for metrizable spaces M , one considers the space M of all mappings from X to M , endowed with the compact-open topology. Since a mapping f : X → X induces a natural mapping M : M → M ′ , an inverse system of compact Hausdorff spaces X determines a direct system M of spaces as well as the corresponding direct system of singular homology groups Hn(M ;G). There is a natural isomorphism between the direct limit dir limHn(M ;G) and the singular homology group Hn(M ;G), where X = inv limX. This continuity theorem, used by some authors, was published more than 50 years ago. Unfortunately, the author discovered a serious error in the proofs of two lemmas on which the result depended. The present paper gives new correct proofs of these lemmas.