Open-constructible functions

We prove that if a continuous one-to-one function between subspaces X, Y of the Cantor set C maps each open set into constructible or, more general, into resolvable one, then f is a piecewise homeomorphism; i.e., X admits a countable cover C consisting of pairwise disjoint closed sets, such that for each C∈C the restriction f|C is a homeomorphism.