Hi all,Let λ>0 and define an equivalence relation onby Jooolz Tags: λ>0, define, equivalence, relation 

#1
Jan1912, 07:21 PM

P: 11

Hi all,
Let λ>0 and define an equivalence relation on ℝ^{n}{0} by (x~y) [itex]\Leftrightarrow[/itex] (there is an s[itex]\in[/itex]Z such that λ^{s}x=y) I would like to know what the quotient space ℝ^{n}{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[itex]\in[/itex] (1, λ] there is an equivalence class [a]. And for b[itex]\in[/itex] (λ, λ^{2}] we have b=rλ where 1< r ≤ λ. But this means that b~r. So b[itex]\in[/itex][r] and we know that r[itex]\in[/itex] (1, λ]. The same idea holds for elements from the intervals (λ^{i}, λ^{i+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 ℝ^{n}{0}/~ is homeomorphic to S^{1} x S^{n1}. I would really appreciate any help or hints, to make me understand better what they mean by quotientspace. Kind regards, J. 


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