Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Scot Adams and Alexander S. Kechris, Linear algebraic groups and countable Borel equivalence relations, J. Amer. Math. Soc. 13 (2000), no. 4, 909–943.
Howard Becker, Some theorems and questions about some natural examples in descriptive set theory, manuscript, 1984.
_____, Simply connected sets and hyperarithmetic paths, manuscript, 1986.
_____, On classifying simple connectedness in the projective hierarchy, manuscript, 1987.
_____, Descriptive set theoretic phenomena in analysis and topology, Set Theory of the Continuum (H. Judah, W. Just, and H. Woodin, eds.), Springer-Verlag, 1992, pp. 1–25.
_____, The number of path-components of a compact subset of R n, Logic Colloquium’ 95 (Haifa), Lecture Notes in Logic, vol. 11, Springer-Verlag, Berlin, 1998, pp. 1–16.
R.H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51.
Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions, Cambridge University Press, Cambridge, 1996.
Howard Becker and Roman Pol, Note on path-components in complete spaces, Topology Appl. 114 (2001), no. 1, 107–114.
Riccardo Camerlo, Universal analytic preorders arising from surjective functions, Fund. Math., 187 (2005), 193–212.
_____, Universality of embeddability relations for coloured total orders, Order 22 (2005), 289–300.
_____, Continua and their σ-ideals, Topology Appl. 150 (2005), 1–18.
Robert Cauty, Tadeusz Dobrowolski, Helma Gladdines, and Jan van Mill, Les hyperespaces des rétractes absolus et des rétractes absolus de voisinage du plan, Fund. Math. 148 (1995), no. 3, 257–282 (French).
Riccardo Camerlo, Udayan B. Darji, and Alberto Marcone, Classification problems in continuum theory, Trans. Amer. Math. Soc. 357 (2005), 4301–4328.
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.
H. Cook and A. Lelek, Weakly confluent mappings and atriodic Suslinian curves, Canad. J. Math. 30 (1978), no. 1, 32–44.
Udayan B. Darji, Complexity of hereditarily decomposable continua, Topology Appl. 103 (2000), no. 3, 243–248.
Udayan B. Darji and Alberto Marcone, Complexity of curves, Fund. Math. 182 (2004), no. 1, 79–93.
Tadeusz Dobrowolski and Leonard R. Rubin, The space of ANR’s in ℝn, Fund. Math. 146 (1994), no. 1, 31–58.
Ryszard Engelking, Dimension theory, North-Holland Publishing Co., Amsterdam, 1978, Translated from the Polish and revised by the author, North-Holland Mathematical Library, 19.
Harvey Friedman and Lee Stanley, A Borel reducibility theory for classes of countable structures, J. Symbolic Logic 54 (1989), no. 3, 894–914.
Su Gao and Alexander S. Kechris, On the classification of Polish metric spaces up to isometry., Mem. Am. Math. Soc. 766 (2003), 78 p.
Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, v. 152, Birkhäuser, Boston, 1999.
Greg Hjorth, Classification and orbit equivalence relations, American Mathematical Society, Providence, RI, 2000.
W.T. Ingram and Howard Cook, A characterization of indecomposable compact continua, Topology Conference (Arizona State Univ., Tempe, Ariz., 1967), Arizona State Univ., Tempe, Ariz., 1968, pp. 168–169.
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, no. 156, Springer-Verlag, New York, 1995.
_____, 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.
Alexander S. Kechris, Alain Louveau, and W. Hugh Woodin, The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), no. 1, 263–288.
Paweł Krupski, Hyperspaces of universal curves and 2-cells are true F σ δ -sets, Colloq. Math. 91 (2002), no. 1, 91–98.
_____, More non-analytic classes of continua, Topology Appl. 127 (2003), no. 3, 299–312.
_____, Families of continua with the property of Kelley, arc continua and curves of pseudo-arcs, Houston J. Math. 30 (2004), no. 2, 459–482.
Paweł Krupski and Alicja Samulewicz, Strongly countable dimensional compacta form the Hurewicz set, Topology Appl., to appear.
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.
Hisao Kato and Xiangdong Ye, On Burgess’s theorem and related problems, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2501–2506.
A. Lelek, On the topology of curves. II, Fund. Math. 70 (1971), no. 2, 131–138.
Alain Louveau and Christian Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357 (2005), 4839–4866.
Stefan Mazurkiewicz, Sur le continus absolutement indécomposables, Fund. Math. 16 (1930), 151–159.
_____, Sur l’ensemble des continus péaniens, Fund. Math. 17 (1931), 273–274.
Alberto Marcone and Christian Rosendal, The complexity of continuous embeddability between dendrites, J. Symbolic Logic 69 (2004), no. 3, 663–673.
Sam B. Nadler, Jr., Continuum theory, Marcel Dekker Inc., New York, 1992.
C. Wayne Proctor, A characterization of hereditarily indecomposable continua, Proc. Amer. Math. Soc. 34 (1972), 287–289.
C. Ryll-Nardzewski, On a Freedman’s problem, Fund. Math. 57 (1965), 273–274.
Christian Rosendal, Cofinal families of Borel equivalence relations and quasiorders, J. Symbolic Logic 70 (2005), 1325–1340.
Alicja Samulewicz, The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set, preprint, 2003.
Jack Segal, Quasi dimension type. II: Types in 1-dimensional spaces, Pac. J. Math. 25 (1968), no. 2, 353–370.
Nikolaos Efstathiou Sofronidis, Natural examples of Π 50 -complete sets in analysis, Proc. Amer. Math. Soc. 130 (2002), 1177–1182.
Sławomir Solecki, Descriptive set theory in topology, Recent progress in general topology, II, North-Holland, Amsterdam, 2002, pp. 485–514.
_____, The space of composants of an indecomposable continuum, Adv. Math. 166 (2002), no. 2, 149–192.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Marcone, A. (2006). Complexity of Sets and Binary Relations in Continuum Theory: A Survey. In: Bagaria, J., Todorcevic, S. (eds) Set Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7692-9_5
Download citation
DOI: https://doi.org/10.1007/3-7643-7692-9_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7691-8
Online ISBN: 978-3-7643-7692-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)