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

  1. 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,
  2. jcsd
  3. micromass

    micromass 18,961
    Staff Emeritus
    Science Advisor
    Education Advisor

    Re: Quotientspace

    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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