On almost one-to-one maps

A continuous map $f:X\to Y$ of topological spaces $X, Y$ is said to be almost $1$-to-$1$ if the set of the points $x\in X$ such that $f^{-1}(f(x))=\{x\}$ is dense in $X$; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and $\sigma$-compact spaces (e.g., $n$-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if $f$ is a minimal self-mapping of a 2-manifold $M$, then point preimages under $f$ are tree-like continua and either $M$ is a union of 2-tori, or $M$ is a union of Klein bottles permuted by $f$.