Editing Mayer–Vietoris sequence (section)

===Naturality===
The homology groups are [[Natural (category theory)|natural]] in the sense that if <math>f:X_1 \to X_2</math> is a [[Continuous function (topology)|continuous]] map, then there is a canonical [[pushforward (homology)|pushforward]] map of homology groups <math>f_*: H_k(X_1) \to H_k(X_2)</math> such that the composition of pushforwards is the pushforward of a composition: that is, <math>(g\circ h)_* = g_*\circ h_*.</math> The Mayer–Vietoris sequence is also natural in the sense that if

:<math>\begin{matrix} X_1 = A_1 \cup B_1 \\ X_2 = A_2 \cup B_2 \end{matrix} \qquad \text{and} \qquad \begin{matrix} f(A_1) \subset A_2 \\f(B_1) \subset B_2\end{matrix}</math>,

then the connecting morphism of the Mayer–Vietoris sequence, <math>\partial_*,</math> commutes with <math>f_*</math>.<ref>{{harvnb|Massey|1984|p=208}}</ref> That is, the following diagram [[Commutative diagram|commutes]]<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.4}}</ref>  (the horizontal maps are the usual ones):

:<math>\begin{matrix}
\cdots & H_{n+1}(X_1) & \longrightarrow & H_n(A_1\cap B_1) & \longrightarrow & H_n(A_1)\oplus H_n(B_1) & \longrightarrow & H_n(X_1) & \longrightarrow &H_{n-1}(A_1\cap B_1) & \longrightarrow & \cdots\\ 
& f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow\\
\cdots & H_{n+1}(X_2) & \longrightarrow & H_n(A_2\cap B_2) & \longrightarrow & H_n(A_2)\oplus H_n(B_2) & \longrightarrow & H_n(X_2) & \longrightarrow &H_{n-1}(A_2\cap B_2) & \longrightarrow & \cdots\\ 
\end{matrix}</math>