Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T08:19:04.668Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPACT SPACES OF THE FIRST BAIRE CLASS THAT HAVE OPEN FINITE DEGREE

Part of: Real functions Fairly general properties Extremal combinatorics Graph theory Connections with other structures, applications Set theory

Published online by Cambridge University Press:  21 November 2016

Antonio Avilés  and
Stevo Todorcevic
Antonio Avilés
Affiliation:
Universidad de Murcia, Departamento de Matemáticas, Campus de Espinardo 30100 Murcia, Spain ([email protected])
Stevo Todorcevic
Affiliation:
Institut de Mathématiques de Jussieu, CNRS UMR 7586, Case 247, 4 place Jussieu, 75252 Paris Cedex, France Department of Mathematics, University of Toronto, Toronto, Canada  M5S 3G3 ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

Keywords

Rosenthal compactfirst Baire class functionmultidimensionalsplit intervalAlexandroff duplicate

MSC classification

Primary: 26A21: Classification of real functions; Baire classification of sets and functions 03E15: Descriptive set theory 05C05: Trees 05D10: Ramsey theory 54D30: Compactness 54D55: Sequential spaces 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 5 , November 2018 , pp. 1173 - 1196
Copyright
© Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

First author supported by MINECO and FEDER (MTM2014-54182-P) and by Fundación Séneca - Región de Murcia (19275/PI/14). Second author partially supported by grants from NSERC and CNRS.

References

Argyros, S., Dodos, P. and Kanellopoulos, V., A classification of separable Rosenthal compacta and its applications, Diss. Math. 449 (2008), 52 pp.Google Scholar
Avilés, A., An introduction to multiple gaps, Zb. Rad. Matematički institut. 17(25) (2015), 7–32.Google Scholar
Avilés, A., Poveda, A. and Todorcevic, S., Rosenthal compacta that are premetric of finite degree, Preprint, arXiv:1512.06070.Google Scholar
Avilés, A. and Todorcevic, S., Finite basis for analytic strong n-gaps, Combinatorica 33(4) (2013), 375–393.Google Scholar
Avilés, A. and Todorcevic, S., Finite basis for analytic n-gaps, Publ. Math. Inst. Hautes Études Sci. 121(1) (2015), 57–79.Google Scholar
Bourgain, J., Fremlin, D. and Talagrand, M., Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100(4) (1978), 845–886.Google Scholar
Debs, G., Descriptive aspects of Rosenthal compacta, in Recent Progress in General Topology, III, pp. 205–227 (Atlantis Press, Paris, 2014).Google Scholar
Kechris, A., Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156 (Springer-Verlag, New York, 1995).Google Scholar
Knaust, H., Array convergence of functions of the first Baire class, Proc. Amer. Math. Soc. 112(2) (1991), 529–532.Google Scholar
Milliken, K. R., A partition theorem for the infinite subsets of a tree, Trans. Amer. Math. Soc. 263 (1981), 137–148.Google Scholar
Odell, E. and Rosenthal, H., A double-dual characterization of separable Banach spaces containing ℓ 1 , Israel J. Math. 20 (1975), 375–384.Google Scholar
Pol, R., Note on pointwise convergence of analytic sets, Mathematika 36 (1989), 290–300.Google Scholar
Rosenthal, H. P., Pointwise compact subsets of first Baire class, Amer. J. Math. 99 (1977), 362–378.Google Scholar
Todorcevic, S., Compact subsets of the first Baire class, J. Amer. Math. Soc. 12 (1999), 1179–1212.Google Scholar
Todorcevic, S., Introduction to Ramsey Spaces, Annals of Mathematics Studies, 174 (Princeton University Press, 2010).Google Scholar