Given $X$ and $X_1 \subseteq X$, $X_2 \subseteq X$ we have the inclusion maps $$i_h: X_h \hookrightarrow X$$
for h=1,2. These inclusions induce the following maps between singular chain groups: $$i_{\#h}: \mathcal{S}_{\bullet}(X_h) \longrightarrow \mathcal{S}_{\bullet}(X)$$
I don't understand why $i_{\#h}$ would be injective, given that $i_{\#h} = i_h \circ \sigma_q$, where $\sigma_q: \Delta_q \rightarrow X_h$ is the singular simplex which needs not to be injective.
Can anyone help me?
It's not about a simplex $\sigma_q$ being injective or not. It's about the linear map $i_{\#h}$ mapping simplices $\neq 0$ in the singular chain group of $X_h$ to ones $\neq 0$ in the singular chain group of $X$. And certainly this is true. Be aware that $i_{\#h} = i_h \circ \sigma_q$ doesn't make sense, it should be $i_{\#h}(\sigma_q) = i_h \circ \sigma_q$.