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Math Help - homeomorphism and projective space

  1. #1
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    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!
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  2. #2
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    Let H be the two-element subgroup of \mathrm{Homeo}\,(S^{n-1}) consisting of maps x\mapsto x and x\mapsto -x. Then P\,\mathbb{R}^{n-1} = S^{n-1}/R_H and the quotient map is 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 f:X\to S^{n-1},\;x\mapsto \frac x{|x|} and g:S^{n-1}\to X by inclusion and look for homeomorphisms of the form p' \circ f \circ q with inverse q'\circ g \circ p.
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  3. #3
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    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?
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  4. #4
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    Since p is open and R_G is closed, it follows that the quotient X/R_G is Hausdorff; see here.
    Last edited by maddas; April 25th 2010 at 03:23 PM.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by maddas View Post
    Since p is open and R_G is closed, it follows that the quotient X/R_G is Hausdorff; see here.
    From this the second problem automatically follows. Namely, since X/R_G is Hausdorff and p:X\to X/R_G continuous then it is clearly true that \Gamma_p is closed.

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