# Algebric geometry

1. Sep 28, 2004

### 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

2. Sep 28, 2004

### Hurkyl

Staff Emeritus
(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?

3. Sep 28, 2004

### 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.

4. Sep 30, 2004

### Feynman

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

5. Sep 30, 2004

### 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.

6. Sep 30, 2004

### 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.

7. Nov 6, 2004

thx

8. Nov 6, 2004

### matt grime

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

9. Nov 10, 2004