Show that canonical projection $\pi:\mathbb{R}^{n+1}\setminus \{0\}\rightarrow \mathbb{RP}^{n}$ is open.
I tried to prove it, but I have a hard time pulling the aberts. I search the internet but only find the projection using the sphere $\pi:\mathbb{S}^{n}\rightarrow \mathbb{RP}^{n}$.
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$\begingroup$$\def\R{\mathbb R}$ $\def\Rnp{\mathbb R^{n+1}}$ Let $U \in \Rnp$ open. We need to show that $\pi^{-1}(\pi(U))$ is open. But $$\pi^{-1}(\pi(U)) = \{r x : r \in \R \setminus \{0\}, x \in U\} = \bigcup_{r\in \R \setminus \{0\}} r U,$$ which is a union of open sets, hence open.
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