Locally one-to-one and covering maps (Q2786843)

A map \(f:X\rightarrow Y\) is said to be locally one-to-one provided that there exists an open cover \(\{ U_\alpha:\alpha\in\mathcal A\}\) of \(X\) such that the restriction \(f|_{U_\alpha}:U_\alpha\rightarrow Y\) is a one-to-one map for each \(\alpha \in\mathcal{A}\). \textit{J. Camargo} [Rev. Colomb. Mat. 45, No. 2, 167--177 (2011; Zbl 1276.54021)] studied necessary and sufficient conditions on a continuum \(X\) such that whenever \(f:X\rightarrow X\) is a locally one-to-one map, we have that \(f\) is a homeomorphism. Every local homeomorphism is a locally one-to-one map and also one easily can see that if a map is locally one-to-one and open, then it is a local homeomorphism. Let \(f:X\rightarrow Y\) be a map between continua. In this paper the authors show that \(f\) is a local homeomorphism if and only if \(f\) is locally one-to-one and there exists a positive integer \(m\) such that \(|f^{-1}(y)|=m\) for each \(y\in Y\). They also prove that every locally one-to-one map between simple closed curves is open. Let \(X\) be a continuum and let \(Y\) be a hereditarily indecomposable continuum such that every proper subcontinuum of \(Y\) is tree-like. The authors show that if \(f:X\rightarrow Y\) is a locally one-to-one map then \(X\) is indecomposable and \(f\) is a local homeomorphism. The authors also study locally one-to-one maps defined between arc-continua, i.e. continua \(X\) such that every proper and nondegenerate subcontinuum of \(X\) is an arc. Moreover, they also give a result that shows a case when homeomorphisms are the only locally one-to-one maps onto arc-continua. The authors also give an example of arc-continua \(X\) and \(Y\) and a locally one-to-one not confluent map \(f:X\rightarrow Y\) such that \(f\) is not a local homeomorphism. They prove that there exists a locally one-to-one map \(f:X\rightarrow Y\) such that \(Y\) is an indecomposable continuum and \(X\) is a decomposable continuum and also such that \(Y\) is an irreducible and unicoherent continuum and \(X\) is a continuum that is neither irreducible nor unicoherent. Then they prove, for continua \(X\) and \(Y\), if there exists a surjective locally one-to-one map \(f:X\rightarrow Y\), then \(\dim(X)=\dim(Y)\).