The question is that 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 set to open set. 

    The only way I know is that one imposes 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.