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Using homotopy it is easy to prove that (in topology) $\mathbb{R}^n\cong \mathbb{R}^m$ if and only if $n=m$. This result seems intuitively true, but, as realized very earlier and almost everyone who tries to prove it, that the proof is not so easy. Here I have two questions:

Question 1: Before invention of homotopy or homology theory, was this proved? who and how?

Question 2: Are there other proofs of this theorem now?

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    It's not an old result, and I think the first proofs already used homotopy. See https://en.wikipedia.org/wiki/Invariance_of_domain – Captain Lama Apr 14 '16 at 09:57
  • I think there are some slightly weaker results which can be proved without homotopical methods at least. Something like (trying to recall from memory so may be inaccurate) "If $n$ is odd and $m$ is even then $\mathbb{R}^n \not\cong \mathbb{R}^m$." And of course the fact that $\mathbb{R}\not\cong\mathbb{R}^n$ for $n\geq 2$ is easily proved using the cut-point-property. – Dan Rust Apr 14 '16 at 10:16
  • Here is a link for the odd/even statement – Dan Rust Apr 14 '16 at 10:22

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