Consider $\mathbb{C} P^n$ and its dual space, which consists of hyperplanes in $\mathbb{C} P^n$. Are they homeomoprhic?
I read this fact somewhere, but can't remember where. Also i don't even remember which topology must we choose for this fact to be correct.
The topology considered is that of Zariski. The map $F : \mathbb{P}^{n} \longrightarrow \ (\mathbb{P^{n}})^{\vee}$ defined by: $$F([a_{0} : \cdots : a_{n}]) = H = `\left\{[z_{0} : \cdots z_{n}] \in \mathbb{P}^{n}; a_{0}z_{0} + \cdots +a_{n}z_{n} = 0\right\}$$ provides the desired homeomorphism. Where $(\mathbb{P^{n}})^{\vee}$ denotes the dual space of $\mathbb{P}^{n}$.
