Singular chain

Let \(X\) be a topological space, \(R\) a commutative ring, and \(p\geq 0\). A singular \(p\)-chain in \(X\) with coefficients in \(R\) is an element of the free \(R\)-module \(C_p(X;R)\) on the singular \(p\)-simplices \(\sigma\colon\Delta^p\to X\), that is, a finite sum \(c = \sum_{i=1}^k r_i\sigma_i\) with \(r_i\in R\) and \(\sigma_i\colon\Delta^p\to X\) continuous.