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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Definable continuous mappings and Whyburn’s conjecture
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by Sĩ Tiệp Dinh and Tien-Son PhẠm
Proc. Amer. Math. Soc. 151 (2023), 2081-2095
DOI: https://doi.org/10.1090/proc/16272
Published electronically: February 10, 2023

Abstract:

For a definable continuous mapping $f$ from a definable connected open subset $\Omega$ of $\mathbb R^n$ into $\mathbb R^n$, we show that the following statements are equivalent:

  1. The mapping $f$ is open.
  2. The fibers of $f$ are finite and the Jacobian of $f$ does not change sign on the set of points at which $f$ is differentiable.
  3. The fibers of ${f}$ are finite and the set of points at which $f$ is not a local homeomorphism has dimension at most $n - 2$.

As an application, we prove that Whyburn’s conjecture is true for definable mappings: A definable surjective open continuous mapping of one closed ball into another which maps boundary homeomorphically onto boundary is necessarily a homeomorphism.

References
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Bibliographic Information
  • Sĩ Tiệp Dinh
  • Affiliation: Institute of Mathematics, VAST, 18, Hoang Quoc Viet Road, Cau Giay District 10307, Hanoi, Vietnam
  • MR Author ID: 885696
  • ORCID: 0000-0001-9116-4534
  • Email: dstiep@math.ac.vn
  • Tien-Son PhẠm
  • Affiliation: Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam
  • MR Author ID: 632781
  • Email: sonpt@dlu.edu.vn
  • Received by editor(s): May 24, 2022
  • Received by editor(s) in revised form: September 6, 2022, September 7, 2022, and September 11, 2022
  • Published electronically: February 10, 2023
  • Additional Notes: This work was supported by the International Centre of Research and Postgraduate Training in Mathematics (ICRTM) under grant number ICRTM01\rule{0.1cm}0.15mm2022.01
    The first author is the corresponding author.
  • Communicated by: Nageswari Shanmugalingam
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 2081-2095
  • MSC (2020): Primary 26B10, 54C10, 03C64
  • DOI: https://doi.org/10.1090/proc/16272
  • MathSciNet review: 4556202