Topological Open Mapping Theorem for mappings between different Euclidean dimensions?

Solution 1:

This was too long for a comment, it is not an answer.

I don't know of any analogues in some arbitrary topological space. here is a list of counter-examples for relatively well-behaved spaces.

One really needs some kind of injectivity requirement, which gives the well-known result: for $X$ compact and $Y$ Hausdorff, a bijective continuous map $f:X \to Y$ is open (and hence a homeomorphism.)

If you add some additional algebraic structure, you could recover the open mapping theorems for functional analysis and sufficiently nice topological groups.

A lukewarm observation: if we let $U \subset \mathbb R^n$ be open, then if $f: U: \to \mathbb R^m$ factors through projection $\pi:\mathbb R^n \to \mathbb R^m$ so that the induced map $\tilde{f}:U \to \mathbb R^m$ is injective, then $f$ is open since it is the composition of open maps.

Hopefully someone else can provide a better answer.