The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ that contains exactly one representative of each equivalence class is easily seen to be an open hemisphere.

Consider now the complex projective $\mathbb{C}P^n$ as the quotient space of $\mathbb{S}^{2n+1} \subset \mathbb{C}^{n+1}$ by the $\mathbb{S}^1$ action given by

$\lambda \cdot (z_1, \dots, z_{n+1}) = (\lambda z_1, \dots, \lambda z_{n+1}), \qquad \lambda \in \mathbb{S}^1, \, (z_1, \dots, z_{n+1}) \in \mathbb{S}^{2n+1}.$

In analogy with the real case, what is now the largest hypersurface of $\mathbb{S}^{2n+1}$ that contains exactly one representative of each equivalence class?