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A fundamental question on homeomorphism 
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#1
Jan1513, 06:35 AM

P: 15

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! 


#2
Jan1513, 07:14 AM

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PF Gold
P: 39,491

Any subspace of R^{n} is R^{k} 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").



#3
Jan1513, 07:26 AM

P: 15

Hi, HallsofIvy,
How about if the subset of R^n is not the whole R^k (k<n) but some illbehaved set (e.g., a space filling line)? 


#4
Jan1513, 08:04 AM

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PF Gold
P: 4,771

A fundamental question on homeomorphism
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. 


#5
Jan1513, 12:46 PM

P: 15

Thank you very much for your helpful answer. It is really a nice proof. 


#6
Jan1913, 03:11 PM

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