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


by Jooolz
Tags: λ>0, define, equivalence, relation
Jooolz
Jooolz is offline
#1
Jan19-12, 07:21 PM
P: 11
Hi all,

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

(x~y) [itex]\Leftrightarrow[/itex] (there is an s[itex]\in[/itex]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[itex]\in[/itex] (1, λ] there is an equivalence class [a]. And for b[itex]\in[/itex] (λ, λ2] 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 (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.
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micromass
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#2
Jan19-12, 09:26 PM
Mentor
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P: 16,698
For elements in ]0,1[, you can show that such elements are also equivalent to an element in [itex][1,\lambda[[/itex]

For elements in [itex]]-\infty,0[[/itex], you can show it equivalent to something in [itex]]-\lambda,-1][/itex].


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