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Homeomorphism and project space

  1. Apr 25, 2010 #1
    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 was pretty straightforward. Could anyone give me some hints how to deal with later ones? Any input is appreciated!
     
  2. jcsd
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