The real projective spaces $\mathbb{R}P^n$:

The projective space $\mathbb{R}P^n$ is the identification space obtained from the sphere $S^n$ by the equivalence relation $\sim$ given by

$${\bf x} \sim {\bf y}$$    if and only if $${\bf x} = -{\bf y},\quad {\bf x}, {\bf y} \in S^n. \eqno(4.2)$$

That is to say, each pair of antipodal points are identified.