Applications of almost one-to-one maps

A continuous map f : X → Y of topological spaces X , Y is said to be almost 1-to-1 if the set of the points x ∈ X such that f −1 ( f ( x ) ) = { x } is dense in X; it is said to be light if pointwise preimages are 0-dimensional. In a previous paper we showed that sometimes almost one-to-one light maps of compact and σ-compact spaces must be homeomorphisms or embeddings. In this paper we introduce a similar notion of an almost d-to-1 map and extend the above results to them and other related maps. In a forthcoming paper we use these results and show that if f is a minimal self-mapping of a 2-manifold 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.