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!