Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. Consider the projectivization of the rank two bundle $\mathcal{O}(-1)\oplus \mathcal{O}$. This is a bundle over $\mathbb{C}\mathbb{P}^\infty$ with fiber $\mathbb{C}\mathbb{P}^1$. 

**How to describe explicitly the cohomology ring of this space?**


Sorry if this question is more appropriate for Mathematics Stack Exchange.