Elementary proof of topological invariance of dimension using Brouwer's fixed point and invariance of domain theorems?

If $V$, open subset of $\Bbb R^m$, were homeomorphic to an open subset of $\Bbb R^n$, $U$, let $f: U \to V$ be a homeomorphism. (Suppose WLOG that $m \leq n$.) Compose with a linear inclusion map $\Bbb R^m \hookrightarrow \Bbb R^n$ to get a continuous injective map $U \to \Bbb R^n$ with image contained in the subspace $\Bbb R^m$, as it factors as $U \to V \subset \Bbb R^m \hookrightarrow \Bbb R^n$.

If $m < n$, the image cannot be open - any neighborhood of a point in the hyperplane contains points not in the hyperplane. Therefore $m=n$. So invariance of domain implies invariance of dimension.

Of course, this says even more: there's not even a continuous injection $U \to V$ between open sets of $\Bbb R^n, \Bbb R^m$ respectively, $n>m$.

Sperner showed in [Sperner 1928] below that invariance of open sets, invariance of domain and invariance of dimension can be proved already with elementary combinatorial methods alone.

They follow simply from the following theorem of Lebesgue:

A bounded point set $G$ in the $n$-dimensional number space is given, which contains inner points. Given a bounded point set $G$ in the $n$-dimensional number space containing inner points. The points of $G$ be distributed to a finite number of closed sets $M_{i}$ ($i=1,2,3,...,s$), so that every point of $G$ occurs at least in one of the sets $M_{i}$. Then there is at least one point in $G$ which lies in at least $n+1$ sets, if only the $M_{i}$ were chosen sufficiently small.

Sperner proved this theorem of Lebesgue by simple combinatorial methods.

The lemma in [Sperner 1928] can also be used in a proof of Brouwer's fixed-point theorem.

[Sperner 1928] Sperner, Emanuel: Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes. In: Abh. Math. Sem. Univ. Hamburg. Band 6, 1928, 265-272

Encyclopedia of Mathematics: Sperner lemma

Encyclopedia of Mathematics: Brouwer theorem