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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On almost one-to-one maps


Authors: Alexander Blokh, Lex Oversteegen and E. D. Tymchatyn
Journal: Trans. Amer. Math. Soc. 358 (2006), 5003-5014
MSC (2000): Primary 57N35, 54C10; Secondary 37B45
Published electronically: June 13, 2006
MathSciNet review: 2231882
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Abstract: 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$.


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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: ablokh@math.uab.edu

Lex Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: overstee@math.uab.edu

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: tymchat@snoopy.math.usask.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03922-5
PII: S 0002-9947(06)03922-5
Keywords: Almost one-to-one map, embedding, homeomorphism, light map
Received by editor(s): February 29, 2004
Received by editor(s) in revised form: October 21, 2004
Published electronically: June 13, 2006
Additional Notes: The first author was partially supported by NSF Grant DMS-0140349
The second author was partially supported by NSF Grant DMS-0072626
The third author was partially supported by NSERC grant OGP005616
Article copyright: © Copyright 2006 American Mathematical Society