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Remark (The Mayer-Vietoris Sequence). Let $X$ be a topological space, let $U,V \subseteq X$ be subsets whose interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$, and set $K = \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }(U \cap V) } \operatorname{Sing}_{\bullet }(V)$. Then the inclusion $K \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ induces a quasi-isomorphism $\mathrm{C}_{\ast }(K; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}})$ (by virtue of Theorem and Proposition, and we have a short exact sequence of chain complexes

\[ 0 \rightarrow \mathrm{C}_{\ast }(U \cap V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{C}_{\ast }(U; \operatorname{\mathbf{Z}}) \oplus \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{C}_{\ast }(K; \operatorname{\mathbf{Z}}) \rightarrow 0. \]

Passing to homology groups (see Construction ), we obtain a long exact sequence of abelian groups

\[ \cdots \rightarrow \mathrm{H}_{\ast +1}(X; \operatorname{\mathbf{Z}}) \xrightarrow {\delta } \mathrm{H}_{\ast }(U \cap V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(U; \operatorname{\mathbf{Z}}) \oplus \mathrm{H}_{\ast }(V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow \cdots \]

which we refer to as the Mayer-Vietoris sequence of the covering $\{ U,V \} $. The existence of this sequence is essentially equivalent to the statement of Theorem