Applications of almost one-to-one maps |
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Authors: | Alexander Blokh Lex Oversteegen ED Tymchatyn |
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Institution: | a Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA b Department of Mathematics, Saskatchewan University, Canada |
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Abstract: | A continuous map 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. |
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Keywords: | primary 54F15 secondary 54C05 |
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