The surjectivity of the canonical homomorphism from singular homology to Cech homology

Abstract

Let X X be a locally n n -connected compact metric space. Then, the canonical homomorphism from the singular homology group H n + 1 ( X ) H_{n+1}(X) to the Čech homology group H ˇ n + 1 ( X ) \check {H}_{n+1}(X) is surjective. Consequently, if a compact metric space X X is locally connected, then the canonical homomorphism from H 1 ( X ) H_1(X) to H ˇ 1 ( X ) {\check H}_1(X) is surjective.