Real projective space
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Contents
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 | |
| [math]\frac{\mathbb{S}^n\subset\mathbb{R}^{n+1} }{\langle x\sim -x\rangle}[/math] |
Definition 2
| Definition 2 | |
| [math]\frac{\mathbb{R}^{n+1}-\{0\} }{\langle x\sim\lambda x\ \vert\ \lambda\in(\mathbb{R}-\{0\})\rangle} [/math] |
- [ilmath]\mathbb{RP}^n:\eq\{L\in\mathcal{P}(\mathbb{R}^{n+1})\ \vert\ (L,\mathbb{R})\text{ is an 1-} [/ilmath][ilmath]\text{dimensional} [/ilmath][ilmath]\text{ vector } [/ilmath][ilmath]\text{subspace} [/ilmath][ilmath]\text{ of }(\mathbb{R}^{n+1},\mathbb{R})\} [/ilmath]
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:
- [math]\mathbb{RP}^n\cong\frac{\mathbb{R}^{n+1}-\{0\} }{\pi} [/math] TODO: What does this actually mean though? In terms of quotient-ing by an equivalence relation!
- [math]\mathbb{RP}^n\cong\frac{\mathbb{R}^{n+1}-\{0\} }{\pi} [/math]
- We use this map to imbue [ilmath]\mathbb{RP}^n[/ilmath] with the quotient topology, so:
Named instances
- Real projective plane - [ilmath]\mathbb{RP}^2[/ilmath]
Standard structure
As a topological [ilmath]n[/ilmath]-manifold
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