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Theorem (Excision for Homology). Let $X$ be a topological space and let $U,V \subseteq X$ be subsets whose interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$. Then the inclusion $U \hookrightarrow X$ induces an isomorphism of relative homology groups

\[ \mathrm{H}_{\ast }( U, U \cap V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(X, V; \operatorname{\mathbf{Z}}). \]

Proof. Let $K$ denote the pushout $\operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }(U \cap V) } \operatorname{Sing}_{\bullet }(V)$. We then have a commutative diagram of short exact sequences of chain complexes

\[ \xymatrix@R =50pt@C=40pt{ 0 \ar [r] & \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \ar@ {=}[d] \ar [r] & \mathrm{C}_{\ast }(K; \operatorname{\mathbf{Z}}) \ar [r] \ar [d]^{\theta '} & \mathrm{C}_{\ast }(U;\operatorname{\mathbf{Z}}) / \mathrm{C}_{\ast }(U \cap V;\operatorname{\mathbf{Z}}) \ar [d]^{\theta } \ar [r] & 0 \\ 0 \ar [r] & \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \ar [r] & \mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}}) \ar [r] & \mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}}) / \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \ar [r] & 0. } \]

Consequently, to show that $\theta $ is a quasi-isomorphism, it will suffice to show that $\theta '$ is a quasi-isomorphism (Remark This is a special case of Proposition, since the inclusion $K \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence of simplicial sets (Theorem $\square$