Skip to main content
×
Home

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

  • VERÓNICA BECHER (a1) and SERGE GRIGORIEFF (a2)
Abstract

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T 0 or T 1, but not T 2 (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.

Copyright
References
Hide All
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.
Bennett H. and Lutzer D. (2009) Strong completeness properties in topology. Questions and Answers in General Topology 27 107–12.
Choquet G. (1969) Lectures on Analysis. Volume I: Integration and Topological Vector Spaces, Benjamin.
de Brecht M. (2013) Quasi-Polish spaces. Annals of Pure and Applied Logic 164 (3) 356381.
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).
Debs G. (1985) Stratégies gagnantes dans certains jeux topologiques. Fundamenta Mathematicae 126 93105.
Dehornoy P. (1986) Turing complexity of the ordinals. Information Processing Letters 23 (4) 167170.
Dorais F. G. and Mummert C. (2010) Stationary and convergent strategies in Choquet games. Fundamenta Mathematicae 209 5979.
Edalat A. (1997) Domains for computation in mathematics, physics and exact real arithmetic. Bulletin of Symbolic Logic 3 (4) 401452.
Ershov Y. (1968) On a hierarchy of sets II. Algebra and Logic 7 (4) 1547.
Galvin F. and Telgárky R. (1986) Stationary strategies in topological games. Topology and its Applications 22 5169.
Gierz G., Hofmann K. H., Keimel K., Lawson J. D., Mislove M. and Scott D. S. (2003) Continuous Lattices and Domains, Cambridge University Press.
Grigorieff S. (1990) Every recursive linear ordering has an isomorphic copy in DTIME-SPACE(n, log(n)). Journal of Symbolic Logic 55 (1) 260276.
Hertling P. (1996a) Unstetigkeitgrade con Funktionen in der effektiven Analysis, Ph.D. thesis, FernUniversity in Hagen.
Hertling P. (1996b) Topological complexity with continuous operations. Journal of Complexity 12 (4) 315338.
Kechris A. S. (1995) Classical Descriptive Set Theory, Springer.
Künzi H.-P. (1983) On strongly quasi-metrizable spaces. Archiv der Mathematik 41 (1) 5763.
Kuratowski K. (1966) Topology, volume I, Academic Press.
Marker D. (2002) Descriptive Set Theory. Lecture Notes. On Marker's home page. Available at http://homepages.math.uic.edu/~marker/math512/dst.ps.
Moschovakis Y. (1979/2009) Descriptive Set Theory, volume 155, American Mathematical Society. (First edition 1979, second edition 2009.)
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 159163.
Schmidt W. W. (1966) On badly approximable numbers and certain games. Transactions of the American Mathematical Society 123 178199.
Selivanov V. L. (2003) Wadge degrees of ω-languages of deterministic Turing machines. Theoretical Informatics and Applications 37 (1) 6783.
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 97108.
Selivanov V. L. (2005) Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly 51 (1) 4561.
Selivanov V. L. (2006) Towards a descriptive set theory for domain-like structures. Theoretical Computer Science 365 (3) 258282.
Selivanov V. L. (2008) On the difference hierarchy in countably based T 0-spaces. Electronic Notes in Theoretical Computer Science 221 257269.
Spector C. (1955) Recursive well-orderings. Journal of Symbolic Logic 20 (2) 151163.
Tang A. (1981) Wadge reducibility and Hausdorff difference hierarchy in . Lectures Notes in Mathematics 871 360371.
Weihrauch K. (2000) Computable Analysis. An Introduction, Springer.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 15 *
Loading metrics...

Abstract views

Total abstract views: 118 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd November 2017. This data will be updated every 24 hours.