On 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 E 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 a-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.