I am trying to apply Mayer-Vietoris to compute the Homology of the complex projective plane $\mathbb{C}P^2$.
What are two nice subsets whose union's interior covers $\mathbb{C}P^2$? I know the CW-structure has three components ($e^0, e^2, e^4$) which may not help.
Would I need to find homology of $\mathbb{C}P^2 / \mathbb{C}P^1 = e^0 \cup e^4$ ?
Consider a small disc $U\cong D^4\subseteq \mathring e^4$ contained inside the interior of the open 4-cell. This gives a covering by the two sets $U$ and $V=(\mathbb{C}P^2-U)$ (technically we'll want to thicken the sets up so that they have a small non-empty intersection, but this isn't a problem since we are working in the interior of an open 4-cell).
Now $\mathbb{C}P^2=U\cup V$ is the pushout of $V\leftarrow U\cap V\rightarrow U$, and we can apply Mayer-Vietoris.
We know the homologies of $U$, $V$, $U\cap V$, since we can easily identify their homotopy types. In particular $U\cong D^4\simeq \ast$, $U\cap V\simeq \partial D^4\simeq S^3$ and $V=(\mathbb{C}P^2-U)\simeq \mathbb{C}P^1\cong S^2 $ by a deformation retraction.
Mayer-Vietoris now gives us a long exact sequence, and there should be no trouble obtaining $H^*\mathbb{C}P^2$ from it.
Observe that what we have really done here is use the pushout diagram that attaches the 4-cell to the 3-skeleton $\mathbb{C}^1\cong S^2$. Our argument thickened everything up to put it in line with classical singular cohomology, but what we really used was that Mayer-Vietoris works for homotopy pushouts, which you may come across in future.