TITLE: compactness of the real projective plane
QUESTION [8 upvotes]: Let $\mathbb{P}_2(\mathbb{R})$ =( $\mathbb{R}^3$-{0})/~  where x~$r$x for any nonzero point x $\in \mathbb{R}^3$ and any nonzero $r \in \mathbb{R}$
I want to show that the $\mathbb{P}_2(\mathbb{R})$ is compact..
I used quotient map to solve, but it doesn't work.  Would you inform me?

REPLY [6 votes]: Hint: $\mathbb{P}^2(\mathbb{R})$ is homeomorphic to the space $S^2/{\sim}$ where $x\sim -x$ for all $x\in S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$. We know $S^2$ is compact by Heine-Borel.