Theorem 4.5:

(i) The projective spaces are compact and connected.

(ii) The projective space $\mathbb{R}P^n$ is homeomorphic to the identification space $(\mathbb{R}^{n+1} - \{0\})/\sim$ where

$${\bf x} \sim {\bf y}$$    if and only if for some $$\lambda \in \mathbb{R},\; {\bf x} = \lambda {\bf y}.$$

(iii) The projective space $\mathbb{R}P^n$ is homeomorphic to the identification space $E^n/\sim$ where ${\bf x} \sim {\bf y}$ if and only if either ${\bf x} = {\bf y}$ or else ${\bf x}, {\bf y} \in S^{n-1}$ and ${\bf x} = -{\bf y}$.