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

[Jooolz]

Hi all,

Let λ>0 and define an equivalence relation on ℝⁿ-{0} by

(x~y) \Leftrightarrow (there is an s\inZ such that λ^sx=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 S^n-1.

I would really appreciate any help or hints, to make me understand better what they mean by quotientspace.

Kind regards,
J.

 

[micromass]

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].