In an exam recently, I was asked to find the minimal number of contractible sets covering $\mathbb{CP}^3$ by considering the cup-product on relative cohomology. Is there nice a way of doing this, either using the proposed approach or some other?
Note: I am aware of (the existence of) the Lusternik–Schnirelmann category, but since this was not part of the curriculum, I doubt that we were supposed to use it.