On proper branched coverings and a question of Vuorinen

We study global injectivity of proper branched coverings from the open Euclidean n$n$-ball onto an open subset of the Euclidean n$n$-space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when n=3$n=3$ or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen.