Editing Mayer–Vietoris sequence (section)

===Derivation===
Consider the [[Homological algebra#Functoriality|long exact sequence associated to]] the [[short exact sequence]]s of chain groups (constituent groups of [[chain complex]]es)

:<math>0 \to C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \to 0</math>,

where α(''x'') = (''x'', −''x''), β(''x'', ''y'') = ''x'' + ''y'', and ''C''<sub>''n''</sub>(''A'' + ''B'') is the chain group consisting of sums of chains in ''A'' and chains in ''B''.<ref name="Hatcher149"/> It is a fact that the singular ''n''-simplices of ''X'' whose images are contained in either ''A'' or ''B'' generate all of the homology group ''H''<sub>''n''</sub>(''X'').<ref>{{harvnb|Hatcher|2002|loc=Proposition 2.21,  p. 119}}</ref> In other words, ''H''<sub>''n''</sub>(''A'' + ''B'') is isomorphic to ''H''<sub>''n''</sub>(''X''). This gives the Mayer–Vietoris sequence for singular homology.

The same computation applied to the short exact sequences of vector spaces of [[differential form]]s

:<math>0\to\Omega^{n}(X)\to\Omega^{n}(U)\oplus\Omega^{n}(V)\to\Omega^{n}(U\cap V)\to 0 </math>

yields the Mayer–Vietoris sequence for de Rham cohomology.<ref>{{harvnb|Bott|Tu|1982|loc=§I.2}}</ref>

From a formal point of view, the Mayer–Vietoris sequence can be derived from the [[Eilenberg–Steenrod axioms]] for [[homology theory|homology theories]] using the [[long exact sequence in homology]].<ref>{{harvnb|Hatcher|2002|p=162}}</ref>