Skip to main content Accesibility Help

Some hierarchies of QCB 0-spaces


We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into , and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.

Hide All
Brattka, V. and Hertling, P. (2002) Topological properties of real number representations. Theoretical Computer Science 284 241257.
de Brecht, M. (2013) Quasi-Polish spaces. Annals of pure and applied logic 164 356381.
de Brecht, M. and Yamamoto, A. (2009) Σ0 α -admissible representations (extended abstract). In: Bauer, A., Hertling, P. and Ko, K.-I (eds.) 6th international conference on computability and complexity in analysis. OASICS 11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
Engelking, R. (1989) General Topology, Heldermann, Berlin.
Ershov, Yu. L. (1974) Maximal and everywhere defined functionals. Algebra and Logic 13 210225.
Escard, M., Lawson, J. and Simpson, A. (2004) Comparing Cartesian closed categories of (Core) compactly generated spaces. Topology and its Applications 143 105145.
Gierz, G., Hoffmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S. (1980) A compendium of Continuous Lattices, Springer, Berlin.
Hyland, J. M. L. (1979) Filter spaces and continuous functionals. Annals of Mathematical Logic 16 101143.
Junnila, H. and Künzi, H.-P. (1998) Characterizations of absolute Fσδ -sets. Chechoslovak Mathematical Journal 48 5564.
Jung, A. (1990) Cartesian closed categories of algebraic CPO's. Theoretical Computer Science 70 233250.
Kechris, A. S. (1995) Classical Descriptive Set Theory, Springer, New York.
Kleene, S. C. (1959) Countable functionals. In: Heyting, A. (ed.) Constructivity in Mathematics, North Holland, Amsterdam 87100.
Kreisel, G. (1959) Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, North Holland, Amsterdam 101128.
Kreitz, C. and Weihrauch, K. (1985) Theory of representations. Theoretical Computer Science 38 3553.
Normann, D. (1980) Recursion on the Countable Functionals. Lecture Notes in Mathematics 811.
Normann, D. (1981) Countable functionals and the projective hierarchy. Journal of Symbolic Logic 46.2 209215.
Normann, D. (1999) The continuous functionals. In: Griffor, E. R. (Ed.) Handbook of Computability Theory, 251–275.
Schröder, M. (2002) Extended admissibility. Theoretical Computer Science 284 519538.
Schröder, M. (2003) Admissible Representations for Continuous Computations, Ph.D. Thesis, FernUniversität Hagen.
Schröder, M. (2009) The sequential topology on is not regular. Mathematical Structures in Computer Science 19 943957.
Selivanov, V. L. (2004) Difference hierarchy in ϕ-spaces. Algebra and Logic 43.4 238248.
Selivanov, V. L. (2005) Variations on the Wadge reducibility. Siberian Advances in Mathematics 15.3 4480.
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 258282.
Selivanov, V. L. (2008) On the difference hierarchy in countably based T 0-spaces. Electronic Notes in Theoretical Computer Science 221 257269.
Selivanov, V. (2013) Total representations. Logical Methods in Computer Science 9 (2) 130. doi: 10.2168/LMCS-9(2:5)2013
Weihrauch, K. (2000) Computable Analysis, 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? *


Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed