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Let $q:S^2\to X$ identify the two points to $p$. Take disjoint small open disks around the original points and let $U$ be their image; $U$ is a contractible wedge of two disks. Let $V=X\setminus \{p\}$, which is homeomorphic to a sphere with two punctures and retracts onto a circle. Then $U\cap V$ consists of two punctured disks, each retracting onto a circle.

In the Mayer–Vietoris sequence, the map $H_1(U\cap V)={\mathbb{Z}}^2\to H_1(U)\oplus H_1(V)=\mathbb{Z}$ has rank one and is surjective: both annular loops map to generators, with a possible orientation sign. Its kernel is $\mathbb{Z}$, so $H_2(X)=\mathbb{Z}$. At degree zero the map ${\mathbb{Z}}^2\to \mathbb{Z}\oplus \mathbb{Z}$ is $(a,b)\mapsto (a+b,-a-b)$, with rank-one kernel. The preceding degree-one map is surjective, so exactness gives $H_1(X)=\mathbb{Z}$. Since $X$ is connected and both pieces have no higher homology,This illustrates homology after identifying two points on a sphere.