Algebraic Geometry: Proofs, Homeomorphism & Compactness

[Feynman]

Hi
we define the projectif space $$P^n \mathbb{R}$$
by the quotient space :$$\mathbb{R}^{n+1}/\sim$$ where:
$$x\sim y\Leftrightarrow x$$ et$$y$$ are colinaires.
my questions are :
1. How we proof that the restiction de $$\sim$$ on $$S^n$$ (where S^n is the sphere on n dimension) identify x and -x?

2. How this projectif reel space is homeomorphe to the quotient of S^n by this identification?

3.How we proof that [tex]P^{n}\mathbb{R}[\tex] is compact?
thanks

 

[Hurkyl]

(1) Doesn't this follow directly from the definition of ~?

(2) Come up with a 1-1 map between them, show it and its inverse are continuous.

(3) S^n is compact, right?

 

[matt grime]

you don't need to directly show the inverse is continuous: it is a cont bijection from a compact space to a hausdorff space, this makes it automatically a homeomorphism i think.

 

[Feynman]

S^n is compact
What do you mean matt grime?
What is hausdorff space?

 

[matt grime]

don't worry, you'll meet them if and when you need them. it is not necessary for this question which can be done quite easily from the basic definitions.

 

[mathwonk]

a space is hausdorff if any two distinct points have disjoint open nbhds. Then for such a space, compact sets are closed. moreover closed sets are always compact. Since it is trivial to show that any continuous map takes compact sets to compact sets, it follows that any continuous map from a compact space to a hausdorff space takes closed sets to closed sets. hence any continuous bijection from a compact space to a hausdorff space is also a closed map, hence has a continuous inverse.

 

[Feynman]

so do you have an article about this subject ?
thx

 

[matt grime]

Yes, any book about point set topology. Kelley is probably the best bet (kelley's general topology).

 

[Astronuc]

Also one can start with -

http://mathworld.wolfram.com/topics/Topology.html

leading to

http://mathworld.wolfram.com/T2-Space.html

http://mathworld.wolfram.com/HausdorffAxioms.html