Given a continuous mapping $f$ between Euclidean domains (or domains in topological manifolds) of the same (topological) dimension, what are the natural assumptions to conclude that $f$ is open? Here, open means that f maps open sets to open sets.

The only way I know is imposing certain regularity assumption on $f$ to conclude that the fiber of $f$ has zero one-dimensional Hausdorff measure, i.e., $H^1(f^{-1}(y))=0$ for all $y$. Then one invokes a result of C. Titus, G. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962) 329–340 to conclude that $f$ is open. Actually, the latter theorem implies that $f$ is both discrete and open. Here discreteness means that the fiber of $f$ is totally disconnected.