\( \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \DeclareMathOperator{\RP}{\mathbf{RP}} \DeclareMathOperator{\C}{C_*} \DeclareMathOperator{\U}{\mathcal{U}} \)

Section1.5Subdivision and Excision

Definition1.5.1Singular chain complexes with respect to open covers.
If \(\U = \{U_i\}_{i\in I}\) is an open cover of \(X\), let \[ C^{\U}_n(X) = \langle e_\sigma : \sigma\colon \Delta^n \to X,\,\im\sigma\subset U_i\text{ for some }i\rangle \subset C_n(X). \] Observe that \(\im \sigma \subset U_i\) implies \(\im \sigma \circ F_j \subset U_i\) and so \(C^{\U}_*\) is a subcomplex of \(C_*\). Let \(H^{\U}_*\) be the homology of this complex, then we have a map \[ i\colon C^{\U}_*(X) \hookrightarrow C_*(X). \]