Real projective space

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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)


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]

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!

Named instances

Standard structure

As a topological [ilmath]n[/ilmath]-manifold

Grade: A*
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