Definition 29.2 (Singular chains):

A singular $n-$simplex in a topological space $X$ is a continuous map $\sigma:\Delta_n\longrightarrow X$. The free abelian group generated by the set of all singular $n-$simplices in $X$ is called the group of singular $n-$chains in $X$. This group is denoted by $S_n(X)$ and a typical element of $S_n(X)$ is thus a formal sum

$$n_1\sigma_1 + n_2\sigma_2 + \dots + n_k\sigma_k,\eqno(29.3)$$

where the coefficients $n_1, n_2, \dots, n_k$ are integers. For convenience we define $S_{-1}(X)$ to be the zero group.

The most important notion in homology theory is the algebraization of the notion of a boundary which applies to arbitrary singular simplices in an arbitrary topological space and not merely polyhedra in Euclidean spaces obtained by gluing together affine simplices. It is precisely this algebraization which provides considerable flexibility towards applications of homology theory.