Lemma 34.5:

(i) Given an open cover ${\cal U}$ of $X$ and a singular simplex $\sigma\in S_p(X)$, there exists a $k \in \mathbb{N}$ such that ${\cal B}^k\sigma \in S_p^{\;{\cal U}}(X)$. In other words each of the simplicies occurring in ${\cal B}^k\sigma$ has its image in one of the open sets of the cover ${\cal U}$.

(ii) If $\sigma$ is a singular $p$ simplex whose image lies in an open set $U \in {\cal U}$ then ${\cal J}\sigma \in S^{\;{\cal U}}_{p+1}(X)$ where ${\cal J}$ is the chain homotopy constructed in theorem (34.3).