Is there a known criterion for a map to be a local homeomorphism or a.e. local homeomorphism?
Of course, we may add some topological assumptions or analytic assumptions for the mapping, e.g. sense-preserving (means that the local index/degree is positive), discrete (means the fiber of the map is totally disconnected), open (means that it maps open set to open set). It seems that proper analytic assumptions are needed as we know that (non-constant) analytic map in the plane are local homeomorphism.