## 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.## References

- Paul Alexandroff,
*Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung*, Math. Ann.**98**(1928), 617–635. - R. H. Bing,
*Concerning hereditarily indecomposable continua*, Pacific J. Math.**1**(1951), 43–51. MR**43451**, DOI 10.2140/pjm.1951.1.43 - Howard Becker and Alexander S. Kechris,
*The descriptive set theory of Polish group actions*, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR**1425877**, DOI 10.1017/CBO9780511735264 - Riccardo Camerlo and Su Gao,
*The completeness of the isomorphism relation for countable Boolean algebras*, Trans. Amer. Math. Soc.**353**(2001), no. 2, 491–518. MR**1804507**, DOI 10.1090/S0002-9947-00-02659-3 - Udayan B. Darji,
*Complexity of hereditarily decomposable continua*, Topology Appl.**103**(2000), no. 3, 243–248. MR**1758437**, DOI 10.1016/S0166-8641(99)00028-0 - Udayan B. Darji and Alberto Marcone,
*Complexity of curves*, Fund. Math.**182**(2004), no. 1, 79–93. MR**2100716**, DOI 10.4064/fm182-1-4 - Harvey Friedman and Lee Stanley,
*A Borel reducibility theory for classes of countable structures*, J. Symbolic Logic**54**(1989), no. 3, 894–914. MR**1011177**, DOI 10.2307/2274750 - Greg Hjorth,
*Classification and orbit equivalence relations*, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000. MR**1725642**, DOI 10.1090/surv/075 - Alexander S. Kechris,
*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597**, DOI 10.1007/978-1-4612-4190-4 - Alexander S. Kechris,
*Actions of Polish groups and classification problems*, Analysis and logic (Mons, 1997) London Math. Soc. Lecture Note Ser., vol. 262, Cambridge Univ. Press, Cambridge, 2002, pp. 115–187. MR**1967835** - Kazimierz Kuratowski,
*Évaluation de la classe borélienne ou projective d’un ensemble de points à l’aide des symboles logiques*, Fund. Math.**17**(1931), 249–272. - K. Kuratowski,
*Topology. Vol. II*, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR**0259835** - Hisao Kato and Xiangdong Ye,
*On Burgess’s theorem and related problems*, Proc. Amer. Math. Soc.**128**(2000), no. 8, 2501–2506. MR**1653402**, DOI 10.1090/S0002-9939-00-05247-3 - Richard Laver,
*On Fraïssé’s order type conjecture*, Ann. of Math. (2)**93**(1971), 89–111. MR**279005**, DOI 10.2307/1970754 - Stefan Mazurkiewicz,
*Sur l’ensemble des continus péaniens*, Fund. Math.**17**(1931), 273–274. - E. C. Milner,
*Basic wqo- and bqo-theory*, Graphs and order (Banff, Alta., 1984) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 147, Reidel, Dordrecht, 1985, pp. 487–502. MR**818505** - Alberto Marcone and Christian Rosendal,
*The complexity of continuous embeddability between dendrites*, J. Symbolic Logic**69**(2004), no. 3, 663–673. MR**2078915**, DOI 10.2178/jsl/1096901760 - Sam B. Nadler Jr.,
*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552** - Sam B. Nadler Jr.,
*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552** - Stephen G. Simpson,
*Bqo-theory and Fraïssé’s conjecture*, Recursive aspects of descriptive set theory, Oxford University Press, New York, 1985, by Richard Mansfield and Galen Weitkamp, pp. 124–138. - R. H. Sorgenfrey,
*Concerning continua irreducible about $n$ points*, Amer. J. Math.**68**(1946), 667–671. MR**17523**, DOI 10.2307/2371790 - J. van Mill,
*Infinite-dimensional topology*, North-Holland Mathematical Library, vol. 43, North-Holland Publishing Co., Amsterdam, 1989. Prerequisites and introduction. MR**977744**

## 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