Difference between revisions of "Real projective space"
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(Created page with "{{Stub page|grade=A*|msg=Would be a great page to have * Demote to grade C once charts and definition 1 is in place ~~~~}} __TOC__ ==Definition== Let {{M|n\in\mathbb{N}_{\ge 1...") |
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==Standard structure== | ==Standard structure== | ||
===As a [[Topological manifold|topological {{n|manifold}}]]=== | ===As a [[Topological manifold|topological {{n|manifold}}]]=== | ||
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| + | {{Definition|Smooth Manifolds|Topological Manifolds|Manifolds}} | ||
Latest revision as of 09:08, 18 February 2017
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Contents
[hide]Definition
Let n∈N≥1 be given. There are 2 common definitions for RPn that we encounter. We will use definition 1 unless otherwise noted throughout the unified mathematics project.
Definition 1
| Definition 1 | |
| Sn⊂Rn+1⟨x∼−x⟩ |
Definition 2
| Definition 2 | |
| Rn+1−{0}⟨x∼λx | λ∈(R−{0})⟩ |
- RPn:={L∈P(Rn+1) | (L,R) is an 1-dimensional vector subspace of (Rn+1,R)}
Of course doesn't tell us what topology to consider RPn with, for that, define the map:
- π:(Rn+1−{0})→RPn given by: π:x↦⟨x⟩
- We use this map to imbue RPn with the quotient topology, so:
- RPn≅Rn+1−{0}π TODO: What does this actually mean though? In terms of quotient-ing by an equivalence relation!
- RPn≅Rn+1−{0}π
- We use this map to imbue RPn with the quotient topology, so:
Named instances
- Real projective plane - RP2
Standard structure
As a topological n-manifold
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Charts
