# Hi all,Let λ>0 and define an equivalence relation on

by Jooolz
Tags: λ>0, define, equivalence, relation
 P: 11 Hi all, Let λ>0 and define an equivalence relation on ℝn-{0} by (x~y) $\Leftrightarrow$ (there is an s$\in$Z such that λsx=y) I would like to know what the quotient space ℝn-{0}/~ looks like. I know that it is a set of equivalence classes. To understand it better I wanted to see how it works for n=1. In that case I found that for all a$\in$ (1, λ] there is an equivalence class [a]. And for b$\in$ (λ, λ2] we have b=rλ where 1< r ≤ λ. But this means that b~r. So b$\in$[r] and we know that r$\in$ (1, λ]. The same idea holds for elements from the intervals (λi, λi+1]. Hence every element from (1, ∞) will be in an equivalence class which has a representative in (1, λ]. Am I seeing this correct? But how can I specify the equivalence classes of the rest of the elements of ℝ-{0}? (since λ > 1) The goal is, (eventually) to show that the quotientspace ℝn-{0}/~ is homeomorphic to S1 x Sn-1. I would really appreciate any help or hints, to make me understand better what they mean by quotientspace. Kind regards, J.
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 For elements in ]0,1[, you can show that such elements are also equivalent to an element in $[1,\lambda[$ For elements in $]-\infty,0[$, you can show it equivalent to something in $]-\lambda,-1]$.