Sufficient conditions for a local homeomorphism to be injective

Abstract

Let U be an open connected subset of R n whose closure, Ū, is compact and whose boundary, ∂ U, has only finitely many components. Assume f: Ū→ R n is a continuous map such that f| U is a local homeomorphism (e.g. f| U has continuous partial derivatives and nonvanishing Jacobian). Under certain hypotheses on ∂ U and f|∂ U, it is proved that f must map U homeomorphically onto its image.