On almost one-to-one maps
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- by Alexander Blokh, Lex Oversteegen and E. D. Tymchatyn PDF
- Trans. Amer. Math. Soc. 358 (2006), 5003-5014 Request permission
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 $f$.References
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Additional Information
- Alexander Blokh
- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- MR Author ID: 196866
- Email: ablokh@math.uab.edu
- Lex Oversteegen
- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- MR Author ID: 134850
- Email: overstee@math.uab.edu
- E. D. Tymchatyn
- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
- MR Author ID: 175580
- Email: tymchat@snoopy.math.usask.ca
- 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 - © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 5003-5014
- MSC (2000): Primary 57N35, 54C10; Secondary 37B45
- DOI: https://doi.org/10.1090/S0002-9947-06-03922-5
- MathSciNet review: 2231882