Singular chain group
= Singular chain group
{title2=$C_q(X;\mathbb Z)$}
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 $\operatorname{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.