Sufficient conditions for a local homeomorphism to be injective (Q1206523)

Let \(U\) be an open connected subset of \(R^ n\) with compact closure \(\overline U\) and whose boundary \(\partial U\) has only finitely many components. The author formulates conditions for a map \(f:\overline U\to R^ n\) such that \(f| U\) is a local homeomorphism to be a homeomorphism onto its image. The conditions are of two types: first, they assume that each of the components of \(\partial U\) has, homologically speaking, some of the properties of a closed orientable \((n-1)\)-dimensional manifold, and second, they control the behavior of \(f|\partial U\) in a natural fashion.