Real projective space

Stub grade: A*

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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 $n\in\mathbb{N}_{\ge 1}$ be given. There are 2 common definitions for $\mathbb{RP}^n$ 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

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

• $\mathbb{RP}^n:\eq\{L\in\mathcal{P}(\mathbb{R}^{n+1})\ \vert\ (L,\mathbb{R})\text{ is an 1-}$$\text{dimensional}$$\text{ vector }$$\text{subspace}$$\text{ of }(\mathbb{R}^{n+1},\mathbb{R})\}$

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

• $\pi:(\mathbb{R}^{n+1}-\{0\})\rightarrow\mathbb{RP}^n$ given by: $\pi:x\mapsto\langle x\rangle$
  • We use this map to imbue $\mathbb{RP}^n$ 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!

Named instances

• Real projective plane - $\mathbb{RP}^2$

Standard structure

As a topological $n$-manifold

Grade: A*

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Charts