Definition 34.3:

Given an open covering ${\cal U}$ of $X$, $S^{\;\cal U}_n(X)$ denotes the subgroup of $S_n(X)$ generated by all the singular simplicies $\sigma:\Delta_n\longrightarrow X$ such that $\sigma(\Delta_n)\subset U_{\sigma}$ for some open set $U_{\sigma}$ in the covering ${\cal U}$. That is to say, $S^{\;\cal U}_n(X)$ is the free abelian group generated by small simplicies, namely those with images contained in one of the open sets in the given covering. It is clear that that the boundary homomorphism $\partial _n$ maps $S^{\;\cal U}_n(X)$ into $S^{\;\cal U}_{n-1}(X)$ and the resulting subcomplex is denoted by $S^{\;\cal U}(X)$. The homology groups of the complex $S^{\;\cal U}(X)$ will be denoted by $H^{\;\cal U}_n(X)$.