Singular chain ($C_n(X)$)

A singular $n$-chain is a finite formal integer linear combination of singular simplices. Its support is contained in a compact subset of the ambient space.

The singular chain group is the free abelian group on all continuous maps ${\Delta }^q\to X$, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying $\mathrm{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.