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Classification problems in continuum theory


Authors: Riccardo Camerlo, Udayan B. Darji and Alberto Marcone
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
Published electronically: June 9, 2005
MathSciNet review: 2156712
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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
Email: camerlo@calvino.polito.it

Udayan B. Darji
Affiliation: Department of Mathematics, 224 Natural Sciences Building, University of Louisville, Louisville, Kentucky 40292
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

DOI: https://doi.org/10.1090/S0002-9947-05-03956-5
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.
Article copyright: © Copyright 2005 American Mathematical Society

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