# Real projective space

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Would be a great page to have
• Demote to grade C once charts and definition 1 is in place Alec (talk) 06:21, 18 February 2017 (UTC)

## Definition

Let [ilmath]n\in\mathbb{N}_{\ge 1} [/ilmath] be given. There are 2 common definitions for [ilmath]\mathbb{RP}^n[/ilmath] that we encounter. We will use definition 1 unless otherwise noted throughout the unified mathematics project.

### Definition 1

 Definition 1 $\frac{\mathbb{S}^n\subset\mathbb{R}^{n+1} }{\langle x\sim -x\rangle}$

### Definition 2

 Definition 2 $\frac{\mathbb{R}^{n+1}-\{0\} }{\langle x\sim\lambda x\ \vert\ \lambda\in(\mathbb{R}-\{0\})\rangle}$

Of course doesn't tell us what topology to consider [ilmath]\mathbb{RP}^n[/ilmath] with, for that, define the map:

• [ilmath]\pi:(\mathbb{R}^{n+1}-\{0\})\rightarrow\mathbb{RP}^n[/ilmath] given by: [ilmath]\pi:x\mapsto\langle x\rangle[/ilmath]
• We use this map to imbue [ilmath]\mathbb{RP}^n[/ilmath] with the quotient topology, so:
• $\mathbb{RP}^n\cong\frac{\mathbb{R}^{n+1}-\{0\} }{\pi}$
TODO: What does this actually mean though? In terms of quotient-ing by an equivalence relation!