I know that $\mathbb R^n$ and $\mathbb R^m$ are not homeomorphic, and from this fact I want to show there is no homeomorphism between any pair of respective open subsets reasoning by contraction. Does it have a simple argument or should I use a specific theorem?
Induced homeomorphism between $\mathbb R^n$ and $\mathbb R^m$ from the hypothetical existence of a homeomorphism between respective open subsets
general-topologyreal-analysis
Best Answer
Of course
(1) If $n \ne m$, then there is no homeomorphism between an open subset of $\mathbb{R}^n$ and an open subset of $\mathbb{R}^m$
implies
(2) If $n \ne m$, then there is no homeomorphism between $\mathbb{R}^n$ and $\mathbb{R}^m$.
I doubt that you can obtain (1) as an immediate corollary of (2). Both results are usually proved using the machinery of algebraic topology, but proving two results by the same method does not mean that there is simple deduction of one from the other.
A nice overview is contained in
https://terrytao.wordpress.com/tag/invariance-of-domain/