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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Classification problems in continuum theory
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by Riccardo Camerlo, Udayan B. Darji and Alberto Marcone PDF
Trans. Amer. Math. Soc. 357 (2005), 4301-4328 Request permission

Abstract:

We study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics. We provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes. For dendrites with finitely many branch points the homeomorphism and quasi-homeomorphism classes coincide, and the minimal quasi-homeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the homeomorphism relation between dendrites is $S_\infty$-universal. It is shown that the classes of trees and graphs are both $\mathrm {D}_{2}({{\boldsymbol \Sigma _{3}^{0}}})$-complete, the class of dendrites is ${{\boldsymbol \Pi _{3}^{0}}}$-complete, and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is ${{\boldsymbol \Pi _{3}^{0}}}$-complete. We also show that if $G$ is a nondegenerate finitely triangulable continuum, then the class of $G$-like continua is ${\boldsymbol \Pi _{2}^{0}}$-complete.
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Additional Information
  • Riccardo Camerlo
  • Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
  • MR Author ID: 663257
  • Email: camerlo@calvino.polito.it
  • Udayan B. Darji
  • Affiliation: Department of Mathematics, 224 Natural Sciences Building, University of Louisville, Louisville, Kentucky 40292
  • MR Author ID: 318780
  • ORCID: 0000-0002-2899-919X
  • Email: ubdarj01@athena.louisville.edu
  • Alberto Marcone
  • Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 208, 33100 Udine, Italy
  • Email: marcone@dimi.uniud.it
  • Received by editor(s): July 23, 2002
  • Published electronically: June 9, 2005
  • Additional Notes: We thank the referee for making valuable suggestions which made the presentation of the paper clearer.
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 4301-4328
  • MSC (2000): Primary 03E15, 54F15, 54H05; Secondary 06A07
  • DOI: https://doi.org/10.1090/S0002-9947-05-03956-5
  • MathSciNet review: 2156712