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

DOI:
https://doi.org/10.1090/S0002-9947-06-03922-5

Published electronically:
June 13, 2006

MathSciNet review:
2231882

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A continuous map of topological spaces is said to be *almost -to-* if the set of the points such that is dense in ; it is said to be *light* if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and -compact spaces (e.g., -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 is a minimal self-mapping of a 2-manifold , then point preimages under are tree-like continua and either is a union of 2-tori, or is a union of Klein bottles permuted by .

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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:
https://doi.org/10.1090/S0002-9947-06-03922-5

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