TITLE: Is there some elementary proof of invariance of domain?
QUESTION [16 upvotes]: 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!

REPLY [4 votes]: 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).