A remark on the double complex of a covering for singular cohomology

Abstract

Given an open covering of a paracompact topological space X , there are two natural ways to construct a map from the cohomology of the nerve of the covering to the cohomology of X . One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the double complex associated to an open covering, and has a more algebraic flavour. In this paper we prove that these two maps coincide.