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A fundamental question on homeomorphism

  1. Jan 15, 2013 #1
    It is well known that there does NOT exist a homeomorphism between R^m and R^n if m>n. My question is whether it is possible to construct a homeomorphism between R^m (as a whole) and a subset of R^n (note that we also suppose that m>n)?

    Intuitively, it is impossible. Is my intuition right? Thank you for your replying in advance!
     
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  3. Jan 15, 2013 #2

    HallsofIvy

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    Any subspace of Rn is Rk for k< n< m. And you have already said "there does NOT exist a homeomorphism between R^m and R^k if m>k" (where I have replaced your "n" with "k").
     
  4. Jan 15, 2013 #3
    Hi, HallsofIvy,

    How about if the subset of R^n is not the whole R^k (k<n) but some ill-behaved set (e.g., a space filling line)?
     
  5. Jan 15, 2013 #4

    quasar987

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    The usual tool for proving the "no homeo thm" is Brouwer's Invariance of Domain theorem:

    http://en.wikipedia.org/wiki/Invariance_of_domain

    It can in the same way be used to answer your question: Assume a homeo btw S (subset of R^n) and R^m exists. Consider R^n as a subset of R^m (say as R^n x {0,...,0}). Then we have a map

    R^m --> S --> R^m

    which is the homeomorphism of R^m with S composed with the inclusion of R^n in R^m. This map is not open since the inclusion of R^n in R^m maps any subset of R^n to a non open subset of R^m. This contradicts Brouwer's invariance of domain theorem.
     
  6. Jan 15, 2013 #5
    Dear quasar987,

    Thank you very much for your helpful answer. It is really a nice proof.
     
  7. Jan 19, 2013 #6

    Bacle2

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