Then "X" is the union "A"∪"B", with "A" and "B" contractible. Also, the intersection "A"∩"B" is homotopy equivalent to "Y". Hence the Mayer–Vietoris sequence yields, for all "n",  The illustration on the right shows the 1-sphere "X" as the suspension of the 0-sphere "Y". Noting in general that the "k"-sphere is the suspension of the ("k" − 1)-sphere, it is easy to derive the homology groups of the "k"-sphere by induction, as above. A relative form of the Mayer–Vietoris sequence also exists. If "Y" ⊂ "X" and is the union of "C" ⊂ "A" and "D" ⊂ "B", then the exact sequence is:  The homology groups are natural in the sense that if formula_19 is a continuous map, then there is a canonical pushforward map of homology groups formula_20 such that the composition of pushforwards is the pushforward of a composition: that is, formula_21 The Mayer–Vietoris sequence is also natural in the sense that if   then the connecting morphism of the Mayer–Vietoris sequence, formula_23 commutes with formula_24. That is, the following diagram commutes (the horizontal maps are the usual ones):  The Mayer–Vietoris long exact sequence for singular cohomology groups with coefficient group "G" is dual to the homological version. It is the following:  where the dimension preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation. As an important special case when "G" is the group of real numbers R and the underlying topological space has the additional structure of a smooth manifold, the Mayer–Vietoris sequence for de Rham cohomology is  where is an open cover of denotes the restriction map, and is the difference. The map formula_28 is defined similarly as the map formula_29 from above. It can be briefly described as follows. For a cohomology class represented by closed form in , express as a difference of forms formula_30 via a partition of unity subordinate to the open cover , for example. The exterior derivative and agree on and therefore together define an form on .