vote up 0 vote down
star

Why is $\mathbb{R}^n$ not homeomorphic to $\mathbb{R}^m$ when $n \neq m$?

flag
1 
 Too localized, it's Invariance of Domain – Charles Siegel Dec 1 at 20:13
2 
 I don't really think this question is appropriate for mathoverflow, because every math BS should know the answer, just as an important piece of mathematical culture. HINT: the tag on this question should be algebraic topology. – Steven Gubkin Dec 1 at 20:14
Rereading my comment, it seems harsher in print than I thought it would when I typed it. I really just meant to say that it isn't a good question for mathoverflow because any answer you could give without knowing some algebraic topology would be an intro to algebraic topology. So I wanted to just let the OP know that algebraic topology is what they want to learn to answer this question. – Steven Gubkin Dec 1 at 21:34

closed as too localized by Reid Barton, Qiaochu Yuan, Ben Webster♦ Dec 1 at 21:05

2 Answers

vote up 6 vote down

If you had a homeomorphism, call it f, then you would have a homeomorphism $f: \mathbb{R}^n - pt \to \mathbb{R}^m-f(pt)$, where $pt$ is a point of $\mathbb{R}^n$. Now compare the homology groups, and you arrive at a contradiction.

link|flag
7 
 Nice answer, David. I think a two-line proof is more constructive than a long explanation of why the question shouldn't be asked. – Georges Elencwajg Dec 1 at 20:37
But this is two lines plus homology and the questioner may not be familiar with this :) – Arminius Dec 1 at 20:41
2 
 I think it's fair to say that if the questioner were aware of, say, singular homology then he wouldn't have asked the question. I think of this as one of the fundamental questions that motivate the elementary tools of algebraic topology. All this raises the question of what it means to be "of interest to professional mathematicians". When is it OK to ask a basic question in an area with which you're unfamiliar? – Henry Wilton Dec 1 at 21:04
4 
 One thing that is not of much interest is lawyering about what is of interest! A one line response such as David's together with a reference, and a link to MO's FAQ, ends up having signal/noise and interest/signal ratios infinitely higher than anything else I can imagine... – Mariano Suárez-Alvarez Dec 1 at 21:07
vote up 3 vote down

Brouwer's Invariance of Domain.

link|flag

Not the answer you're looking for? Browse other questions tagged or ask your own question.