= Homology after identifying two points on a sphere
{title2=$H_1\cong H_2\cong\mathbb Z$}
Identifying two distinct points of $S^2$ produces a space with <homology groups> $H_0=H_1=H_2=\mathbb Z$ and all higher groups zero. Cover the identified point by the image of two small disks, a contractible wedge of disks, and take as the other open set the complement of the point, a cylinder. Their intersection is two annuli. In the <Mayer–Vietoris sequence>, $H_1$ of this intersection maps onto $H_1$ of the cylinder with a rank-one kernel, producing $H_2=\mathbb Z$. The kernel at $H_0$ has rank one and produces $H_1=\mathbb Z$.
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