Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T07:48:17.577Z Has data issue: false hasContentIssue false
  • English
  • Français

Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization

Published online by Cambridge University Press:  14 November 2014

VERÓNICA BECHER  and
SERGE GRIGORIEFF
VERÓNICA BECHER
Affiliation:
FCEyN, Universidad de Buenos Aires & CONICET, Buenos Aires, Argentina Email: vbecher@dc.uba.ar
SERGE GRIGORIEFF
Affiliation:
LIAFA, CNRS & Université Paris Diderot - Paris 7, France Email: seg@liafa.univ-paris-diderot.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Selivanov in a series of papers centred on algebraic domains. And recently it has been considered by de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper, we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 7: Computing with Infinite Data: Topological and Logical Foundations Part 1 , October 2015 , pp. 1490 - 1519
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, Dov M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, volume 3, Oxford University Press.Google Scholar
Bennett, H. and Lutzer, D. (2009) Strong completeness properties in topology. Questions and Answers in General Topology 27 107–12.Google Scholar
Choquet, G. (1969) Lectures on Analysis. Volume I: Integration and Topological Vector Spaces, Benjamin.Google Scholar
de Brecht, M. (2013) Quasi-Polish spaces. Annals of Pure and Applied Logic 164 (3) 356–381.Google Scholar
Debs, G. (1984) An example of an α-favourable topological space with no α-winning tactic. Séminaire d'Initiation a l'Analyse (Choquet-Rogalski-Saint Raymond).Google Scholar
Debs, G. (1985) Stratégies gagnantes dans certains jeux topologiques. Fundamenta Mathematicae 126 93–105.Google Scholar
Dehornoy, P. (1986) Turing complexity of the ordinals. Information Processing Letters 23 (4) 167–170.Google Scholar
Dorais, F. G. and Mummert, C. (2010) Stationary and convergent strategies in Choquet games. Fundamenta Mathematicae 209 59–79.Google Scholar
Edalat, A. (1997) Domains for computation in mathematics, physics and exact real arithmetic. Bulletin of Symbolic Logic 3 (4) 401–452.Google Scholar
Ershov, Y. (1968) On a hierarchy of sets II. Algebra and Logic 7 (4) 15–47.Google Scholar
Galvin, F. and Telgárky, R. (1986) Stationary strategies in topological games. Topology and its Applications 22 51–69.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous Lattices and Domains, Cambridge University Press.Google Scholar
Grigorieff, S. (1990) Every recursive linear ordering has an isomorphic copy in DTIME-SPACE(n, log(n)). Journal of Symbolic Logic 55 (1) 260–276.Google Scholar
Hertling, P. (1996a) Unstetigkeitgrade con Funktionen in der effektiven Analysis, Ph.D. thesis, FernUniversity in Hagen.Google Scholar
Hertling, P. (1996b) Topological complexity with continuous operations. Journal of Complexity 12 (4) 315–338.CrossRefGoogle Scholar
Kechris, A. S. (1995) Classical Descriptive Set Theory, Springer.CrossRefGoogle Scholar
Künzi, H.-P. (1983) On strongly quasi-metrizable spaces. Archiv der Mathematik 41 (1) 57–63.Google Scholar
Kuratowski, K. (1966) Topology, volume I, Academic Press.Google Scholar
Marker, D. (2002) Descriptive Set Theory. Lecture Notes. On Marker's home page. Available at http://homepages.math.uic.edu/~marker/math512/dst.ps.Google Scholar
Moschovakis, Y. (1979/2009) Descriptive Set Theory, volume 155, American Mathematical Society. (First edition 1979, second edition 2009.)Google Scholar
Oxtoby, J. C. (1957) The Banach–Mazur game and Banach category theorem. In: Contributions to the theory of games, volume III; Annals of Mathematics Studies 39 159–163.Google Scholar
Schmidt, W. W. (1966) On badly approximable numbers and certain games. Transactions of the American Mathematical Society 123 178–199.Google Scholar
Selivanov, V. L. (2003) Wadge degrees of ω-languages of deterministic Turing machines. Theoretical Informatics and Applications 37 (1) 67–83.Google Scholar
Selivanov, V. L. (2003) Wadge degrees of ω-languages of deterministic Turing machines. In: Extended abstract in STACS 2003 Proceedings. Lecture Notes in Computer Science 2607 97–108.Google Scholar
Selivanov, V. L. (2005) Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly 51 (1) 45–61.Google Scholar
Selivanov, V. L. (2006) Towards a descriptive set theory for domain-like structures. Theoretical Computer Science 365 (3) 258–282.CrossRefGoogle Scholar
Selivanov, V. L. (2008) On the difference hierarchy in countably based T 0-spaces. Electronic Notes in Theoretical Computer Science 221 257–269.Google Scholar
Spector, C. (1955) Recursive well-orderings. Journal of Symbolic Logic 20 (2) 151–163.Google Scholar
Tang, A. (1981) Wadge reducibility and Hausdorff difference hierarchy in Pω . Lectures Notes in Mathematics 871 360–371.Google Scholar
Weihrauch, K. (2000) Computable Analysis. An Introduction, Springer.CrossRefGoogle Scholar