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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Characterizing indecomposable plane continua from their complements
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by Clinton P. Curry, John C. Mayer and E. D. Tymchatyn PDF
Proc. Amer. Math. Soc. 136 (2008), 4045-4055 Request permission

Abstract:

We show that a plane continuum $X$ is indecomposable iff $X$ has a sequence $(U_n)_{n=1}^\infty$ of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence $(A_n)_{n=1}^\infty$ of open arcs, with $A_n \subset U_n$ and $\overline {A_n}\setminus A_n \subset \partial U_n$, there is a sequence of shadows $(S_n)_{n=1}^\infty$, where each $S_n$ is a shadow of $A_n$, such that $\lim S_n=X$. Such an open arc divides $U_n$ into disjoint subdomains $V_{n,1}$ and $V_{n,2}$, and a shadow (of $A_n$) is one of the sets $\partial V_{n,i}\cap \partial U$.
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Additional Information
  • Clinton P. Curry
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
  • Email: clintonc@uab.edu
  • John C. Mayer
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
  • Email: mayer@math.uab.edu
  • E. D. Tymchatyn
  • Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0
  • MR Author ID: 175580
  • Email: tymchat@math.usask.ca
  • Received by editor(s): September 4, 2007
  • Published electronically: June 26, 2008
  • Additional Notes: The third author was supported in part by NSERC 0GP005616. We thank the Department of Mathematics and Computer Science at Nipissing University, North Bay, Ontario, for the opportunity to work on this paper in pleasant surroundings at their annual topology workshop.
  • Communicated by: Alexander N. Dranishnikov
  • © Copyright 2008 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 4045-4055
  • MSC (2000): Primary 54F15; Secondary 37F20
  • DOI: https://doi.org/10.1090/S0002-9939-08-09508-7
  • MathSciNet review: 2425746