[proofplan] We derive the Mayer--Vietoris sequence from the [Snake Lemma](/theorems/1930) applied to a short exact sequence of chain complexes. The chain complexes are $C_\bullet(A \cap B)$, $C_\bullet(A) \oplus C_\bullet(B)$, and $C_\bullet^{\mathcal{U}}(X)$ (the subcomplex of $\mathcal{U}$-small chains, where $\mathcal{U} = \{A, B\}$). The maps are the inclusion $(\sigma \mapsto (\sigma, \sigma))$ and the difference $((\alpha, \beta) \mapsto j_{A\#}(\alpha) - j_{B\#}(\beta))$. Exactness of this short exact sequence is verified directly. The Snake Lemma produces a long exact sequence involving $H_n^{\mathcal{U}}(X)$, and the [Small Simplices Theorem](/theorems/2255) identifies $H_n^{\mathcal{U}}(X) \cong H_n(X)$, yielding the Mayer--Vietoris sequence. Naturality follows from the naturality of all the ingredients. [/proofplan] [step:Construct the short exact sequence of chain complexes] Let $\mathcal{U} = \{A, B\}$ be the open cover of $X$, and let $C_n^{\mathcal{U}}(X)$ be the subgroup of $C_n(X)$ generated by singular simplices whose image is contained in $A$ or in $B$. Define chain maps \begin{align*} \Phi_n: C_n(A \cap B) &\to C_n(A) \oplus C_n(B) \\ \sigma &\mapsto \bigl( (i_A)_\#(\sigma),\; (i_B)_\#(\sigma) \bigr) \end{align*} and \begin{align*} \Psi_n: C_n(A) \oplus C_n(B) &\to C_n^{\mathcal{U}}(X) \\ (\alpha, \beta) &\mapsto (j_A)_\#(\alpha) - (j_B)_\#(\beta), \end{align*} where $i_A: A \cap B \hookrightarrow A$, $i_B: A \cap B \hookrightarrow B$, $j_A: A \hookrightarrow X$, $j_B: B \hookrightarrow X$ are the inclusions. Both $\Phi$ and $\Psi$ are chain maps: $\Phi$ commutes with the boundary because $(i_A)_\#$ and $(i_B)_\#$ are chain maps (they are induced by continuous maps), and $\Psi$ commutes with the boundary for the same reason (it is a difference of chain maps). [guided] The setup is to organise the chain groups associated to the open cover $\{A, B\}$ into a short exact sequence. The three chain complexes involved are: - $C_\bullet(A \cap B)$: chains on the intersection. - $C_\bullet(A) \oplus C_\bullet(B)$: chains on each piece, taken together. - $C_\bullet^{\mathcal{U}}(X)$: chains on $X$ that are "small" with respect to the cover (each simplex lands in $A$ or in $B$). The first map $\Phi$ includes a chain on $A \cap B$ into both summands. The second map $\Psi$ takes the difference of chains pushed into $X$. The minus sign in $\Psi$ is essential: without it, the composition $\Psi \circ \Phi$ would not be zero. Why do we use $C_\bullet^{\mathcal{U}}(X)$ instead of $C_\bullet(X)$? Because $\Psi$ is surjective onto $C_\bullet^{\mathcal{U}}(X)$ (every $\mathcal{U}$-small simplex is in $A$ or $B$) but is not surjective onto all of $C_\bullet(X)$ (a simplex whose image meets both $A \setminus B$ and $B \setminus A$ is not in the image of $\Psi$). We will later invoke the [Small Simplices Theorem](/theorems/2255) to replace $H_n^{\mathcal{U}}(X)$ with $H_n(X)$. [/guided] [/step] [step:Verify exactness of $0 \to C_\bullet(A \cap B) \xrightarrow{\Phi} C_\bullet(A) \oplus C_\bullet(B) \xrightarrow{\Psi} C_\bullet^{\mathcal{U}}(X) \to 0$] **Injectivity of $\Phi$:** If $\Phi_n(\sigma) = (0, 0)$, then $(i_A)_\#(\sigma) = 0$ in $C_n(A)$. Since $i_A$ is an inclusion, $(i_A)_\#$ is injective on chains (it is the identity on the underlying simplices, viewed as maps into $A$ rather than $A \cap B$). Hence $\sigma = 0$. **$\operatorname{im}(\Phi) = \ker(\Psi)$:** For any $\sigma \in C_n(A \cap B)$, $\Psi(\Phi(\sigma)) = (j_A)_\#((i_A)_\#(\sigma)) - (j_B)_\#((i_B)_\#(\sigma)) = (j_A \circ i_A)_\#(\sigma) - (j_B \circ i_B)_\#(\sigma) = 0$, since $j_A \circ i_A = j_B \circ i_B$ (both are the inclusion $A \cap B \hookrightarrow X$). So $\operatorname{im}(\Phi) \subset \ker(\Psi)$. Conversely, suppose $\Psi_n(\alpha, \beta) = 0$, i.e., $(j_A)_\#(\alpha) = (j_B)_\#(\beta)$ in $C_n^{\mathcal{U}}(X)$. Since $C_n(A)$ and $C_n(B)$ are free abelian groups on the singular simplices in $A$ and $B$ respectively, the equality $(j_A)_\#(\alpha) = (j_B)_\#(\beta)$ forces every simplex appearing in $\alpha$ (resp. $\beta$) to have image contained in both $A$ and $B$, hence in $A \cap B$. So $\alpha = (i_A)_\#(\gamma)$ and $\beta = (i_B)_\#(\gamma)$ for $\gamma = \alpha$ viewed as a chain in $C_n(A \cap B)$, giving $(\alpha, \beta) = \Phi(\gamma)$. **Surjectivity of $\Psi$:** Every generator of $C_n^{\mathcal{U}}(X)$ is a singular simplex $\sigma: \Delta^n \to X$ with $\sigma(\Delta^n) \subset A$ or $\sigma(\Delta^n) \subset B$. In the first case, $\sigma = \Psi(\sigma, 0)$; in the second, $\sigma = \Psi(0, -\sigma)$. [/step] [step:Apply the Snake Lemma to obtain the long exact sequence with $H_n^{\mathcal{U}}(X)$] The short exact sequence of chain complexes \begin{align*} 0 \to C_\bullet(A \cap B) \xrightarrow{\Phi} C_\bullet(A) \oplus C_\bullet(B) \xrightarrow{\Psi} C_\bullet^{\mathcal{U}}(X) \to 0 \end{align*} satisfies the hypotheses of the [Snake Lemma](/theorems/1930). The Snake Lemma produces connecting homomorphisms $\partial_*: H_n^{\mathcal{U}}(X) \to H_{n-1}(A \cap B)$ and a long exact sequence \begin{align*} \cdots \to H_n(A \cap B) \xrightarrow{\Phi_*} H_n(A) \oplus H_n(B) \xrightarrow{\Psi_*} H_n^{\mathcal{U}}(X) \xrightarrow{\partial_*} H_{n-1}(A \cap B) \to \cdots \end{align*} The maps on homology are $\Phi_* = ((i_A)_*, (i_B)_*)$ and $\Psi_* = (j_A)_* - (j_B)_*$. [/step] [step:Replace $H_n^{\mathcal{U}}(X)$ with $H_n(X)$ via the Small Simplices Theorem] By the [Small Simplices Theorem](/theorems/2255), the inclusion $C_\bullet^{\mathcal{U}}(X) \hookrightarrow C_\bullet(X)$ induces an isomorphism $H_n^{\mathcal{U}}(X) \xrightarrow{\sim} H_n(X)$ for all $n$. Composing $\Psi_*$ with this isomorphism and $\partial_*$ with its inverse, we define $\partial_{MV}: H_n(X) \to H_{n-1}(A \cap B)$ by \begin{align*} \partial_{MV} = \partial_* \circ (\text{inclusion isomorphism})^{-1}. \end{align*} The resulting long exact sequence is \begin{align*} \cdots \to H_n(A \cap B) \xrightarrow{((i_A)_*, (i_B)_*)} H_n(A) \oplus H_n(B) \xrightarrow{(j_A)_* - (j_B)_*} H_n(X) \xrightarrow{\partial_{MV}} H_{n-1}(A \cap B) \to \cdots \end{align*} terminating with $H_0(X) \to 0$. [guided] The Small Simplices Theorem is the key technical ingredient that allows us to pass from "small" chains to all chains. The theorem states that every singular cycle can be subdivided (via iterated barycentric subdivision) until all its simplices are $\mathcal{U}$-small, and this subdivision is chain homotopic to the identity. Therefore $H_n^{\mathcal{U}}(X) \to H_n(X)$ is an isomorphism. Without this theorem, we would only have a long exact sequence involving $H_n^{\mathcal{U}}(X)$, which is a priori a different group from $H_n(X)$. The Small Simplices Theorem identifies them, upgrading the algebraic exact sequence into the geometric Mayer--Vietoris sequence. [/guided] [/step] [step:Verify naturality] Let $f: X \to Y$ be a continuous map with $f(A) \subset U$ and $f(B) \subset V$ for an open cover $Y = U \cup V$. Then $f$ restricts to maps $f|_A: A \to U$, $f|_B: B \to V$, and $f|_{A \cap B}: A \cap B \to U \cap V$. These restrictions induce a morphism of short exact sequences: the diagram \begin{align*} 0 \to C_\bullet(A \cap B) \xrightarrow{\Phi} C_\bullet(A) \oplus C_\bullet(B) \xrightarrow{\Psi} C_\bullet^{\{A,B\}}(X) \to 0 \\ 0 \to C_\bullet(U \cap V) \xrightarrow{\Phi'} C_\bullet(U) \oplus C_\bullet(V) \xrightarrow{\Psi'} C_\bullet^{\{U,V\}}(Y) \to 0 \end{align*} commutes with vertical maps $(f|_{A \cap B})_\#$, $(f|_A)_\# \oplus (f|_B)_\#$, and $f_\#$ (restricted to $\mathcal{U}$-small chains). The naturality of the Snake Lemma's connecting homomorphism, together with the naturality of the Small Simplices isomorphism (the inclusion $C_\bullet^{\mathcal{U}} \hookrightarrow C_\bullet$ is natural in the pair $(X, \mathcal{U})$), gives the commutativity of the Mayer--Vietoris ladder. [/step]