Canonical projection is open - Projective Space

Question

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 $\pi:\mathbb{S}^{n}\rightarrow \mathbb{RP}^{n}$.

You wrote "$\mathbb{R}^{n+1}\backslash$ {0}" Then someone corrected that so that it said "$\mathbb{R}^{n+1}\backslash {0}$". Then I corrected that further so it said "$\mathbb{R}^{n+1}\setminus {0}.$ So note proper MathJax usage. — Michael Hardy, Apr 07 '18 at 17:32
@freegoodman In portuguese, "open set" translates as "conjunto aberto". I'd guess OP is not used to certain math terms in English. — Ivo Terek, Apr 07 '18 at 17:34
I've never seen the word "abert" before and Google isn't helping with that. — Michael Hardy, Apr 07 '18 at 17:35

score 2 · Accepted Answer · edited Apr 07 '18 at 17:32

2

$\def\R{\mathbb R}$ $\def\Rnp{\mathbb R^{n+1}}$ 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.

edited Apr 07 '18 at 17:32

Ivo Terek

80,301

answered Apr 07 '18 at 17:31

fred goodman

4,443

Canonical projection is open - Projective Space

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