# homeomorphism and projective space

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• Apr 25th 2010, 05:23 AM
rain07
homeomorphism and projective space
1. (1) (a) Let X be a topological space. Prove that the set Homeo(X) of home-
omorphisms f : X → X becomes a group when endowed with the binary operation f ◦ g.
(b) Let G be a subgroup of Homeo(X). Prove that the relation ‘xRG y ⇔ ∃g ∈ G such
that g(x) = y ’ is an equivalence relation.
(c) Let G and RG be as in (b), and let p : X → X/RG be the quotient map. Prove that for every U open in X , p(U ) is open in X/RG .
2.(i) Let X = Rn \ {O} , n ≥ 2 , and let G be the subgroup of Homeo(X) composed of the
maps gλ (x) = λx. Prove that X/RG is the real projective space P Rn−1 .
(ii) Prove that the graph of the relation RG described in (i) is closed.
(iii) Prove that P Rn−1 is Hausdorﬀ.

The setup in problem 1 was pretty straightforward. Could anyone give me some hints how to deal with later ones? Any input is appreciated!
• Apr 25th 2010, 10:32 AM
maddas
Let H be the two-element subgroup of $\displaystyle \mathrm{Homeo}\,(S^{n-1})$ consisting of maps $\displaystyle x\mapsto x$ and $\displaystyle x\mapsto -x$. Then $\displaystyle P\,\mathbb{R}^{n-1} = S^{n-1}/R_H$ and the quotient map is $\displaystyle q:S^{n-1}\to S^{n-1}/R_H$. You can define continuous right inverses p' and q' to p and q respectively by chusing a representative for each equivalence class. Now define maps $\displaystyle f:X\to S^{n-1},\;x\mapsto \frac x{|x|}$ and $\displaystyle g:S^{n-1}\to X$ by inclusion and look for homeomorphisms of the form $\displaystyle p' \circ f \circ q$ with inverse $\displaystyle q'\circ g \circ p$.
• Apr 25th 2010, 01:45 PM
rain07
thx
thanks for your reply. But I'm not quite sure that I understand them completely. Could you explain a bit more how to show that Rg is closed and PR^n-1 is Hausdorff?
• Apr 25th 2010, 01:58 PM
maddas
Since p is open and $\displaystyle R_G$ is closed, it follows that the quotient $\displaystyle X/R_G$ is Hausdorff; see here.
• Apr 25th 2010, 08:02 PM
Drexel28
Quote:

Originally Posted by maddas
Since p is open and $\displaystyle R_G$ is closed, it follows that the quotient $\displaystyle X/R_G$ is Hausdorff; see here.

From this the second problem automatically follows. Namely, since $\displaystyle X/R_G$ is Hausdorff and $\displaystyle p:X\to X/R_G$ continuous then it is clearly true that $\displaystyle \Gamma_p$ is closed.

Think about the mapping $\displaystyle p\times\text{id}:X\times R_G\to R_G\times R_G:(x,y)\mapsto (p(x),y)$. This is evidnently continuous and $\displaystyle \Gamma_p=\left(p\times\text{id}\right)^{-1}\left(\Delta_{R_G}\right)$