A Mayer-Vietoris Sequence is a powerful tool used in finding Homology groups for spaces that can be expressed as the unions of simpler spaces from the perspective of Homology theory.
Consider the cover of formed by 2-discs A and B in the figure.
The space is homotopy equivalent to the circle. We know that the homology groups are preserved by homotopy and so for and . Also note how the homology groups of A and B are trivial since they are both contractable. So we know that
This means that since is an isomorphism by exactness.
Consider the cover of the torus by 2 open ended cylinders A and B.