Invariance of domain at least in statement seems a simple result. I mean, the first time I saw the statement I thought: "the proof can't be that bad", but when I searched for it I saw that it needs even algebraic topology to prove this result.
My doubt is: isn't there any other proof of invariance of domain that don't need to use algebraic topology? Is there some more elementary proof of this result?
Thanks very much in advance!
Best Answer
The point here is what means elementary. Many think that a bit of singular homology is not that bad (see Greenberg-Harper's Algebraic Topology textbook, for instance). One can use Mapping Degree Theory as in many books devoted to the topic and use Borsuk-Ulam or else. But I would recommend a carefull reading of
http://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/
There one finds a very nice proof. The heavy part is Brouwer Fixed Point Theorem. Apart from that, some manipulations involving Weierstrass Approximation Theorem and the so-called Little Sard Theorem (in this particular case: a polynomial image of $\mathbb{S}^{n-1}$ in $\mathbb{R}^n$ has empty interior, this disguised in measure zero terms).