Elementary proof of topological invariance of dimension using Brouwer's fixed point and invariance of domain theorems? http://people.math.sc.edu/howard/Notes/brouwer.pdf https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/
These papers give fairly elementary proofs of Brouwer's fixed point and invariance of domain theorems. Having established these tools, is it possible to prove that an open subset of $ \mathbb{R}^n $ and an open subset of $ \mathbb{R}^m $ can't be homeomorphic unless $ n = m $ without refering to all those more advanced things which are normally used here, such as homology theory and algebraic topology?

The following 3 page note proves that open sets in Euclidean spaces are homeomorphic only if their dimensions agree. 
http://personal.colby.edu/~sataylor/teaching/S09/MA331/InvarianceOfDomain.pdf
It uses Sperner's lemma and the notion of topological dimension (or Lebesgue covering dimension: https://en.wikipedia.org/wiki/Lebesgue_covering_dimension), which is defined using open covers, and is a topological invariant, by definition.
