Skip to main content
    • Aa
    • Aa

Wadge-like reducibilities on arbitrary quasi-Polish spaces


The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. Andretta (2006) More on Wadge determinacy. Annals of Pure and Applied Logic 144 (1) 232.

A. Andretta and D. A. Martin (2003) Borel-Wadge degrees. Fundamenta Mathematicae 177 (2) 175192.

M. de Brecht (2013) Quasi-Polish spaces. Annals of Pure and applied Logic 164 356381.

G. Giertz , K. H. Hoffmann , K. Keimel , J. D. Lawson , M. W. Mislove and D. S. Scott (2003) Continuous Lattices and Domains, Cambridge.

O. V. Kudinov and V. L. Selivanov (2007) Definability in the homomorphic quasiorder of finite labelled forests. In: Computation and Logic in the Real World, Springer Lecture Notes in Computer Science 4497 436445.

O. V. Kudinov and V. L. Selivanov (2009) A Gandy theorem for abstract structures and applications to first-order definability. In: Mathematical Theory and Computational Practice, Springer Lecture Notes in Computer Science 5635 290299.

O. V. Kudinov , V. L. Selivanov and A. V. Zhukov (2009) Definability in the h-quasiorder of labelled forests. Annals of Pure and Applied Logic 159 (3) 318332.

O. V. Kudinov , V. L. Selivanov and A. V. Zhukov (2010) Undecidability in Weihrauch degrees. In: Programs, Proofs, Processes. Springer Lecture Notes in Computer Science 6158 256265.

L. Motto Ros (2010b) Beyond Borel-amenability: Scales and superamenable reducibilities. Annals of Pure and Applied Logic 161 (7) 829836.

L. Motto Ros (2011) Game representations of classes of piecewise definable functions. Mathematical Logic Quarterly 57 (1) 95112.

L. Motto Ros (2013) On the structure of finite level and omega-decomposable Borel functions. Journal of Symbolic Logic 78 (4) 12571287.

A. Ostrovsky (2011) σ-homogeneity of Borel sets. Archive for Mathematical Logic 50 (5–6) 661664.

J. Pawlikowski and M. Sabok (2012) Decomposing Borel functions and structure at finite levels of the Baire hierarchy. Annals of Pure and Applied Logic 163 (12) 17481764.

V. L. Selivanov (2004) Difference hierarchy in ϕ-spaces. Algebra Logika 43 (4) 425444 (Engl. Trans.: Algebra Logic 43 (4) 238–248).

V. L. Selivanov (2007) Hierarchies of Δ0 2-measurable k-partitions. Mathematical Logic Quarterly 53 (4–5) 446461.

V. L. Selivanov (2008a) On the difference hierarchy in countably based T 0-spaces. Electronic Notes in Theoretical Computer Science 221 257269.

V. L. Selivanov (2008b) Wadge reducibility and infinite computations. Mathematics in Computer Science 2 536.

V. L. Selivanov (2010) On the Wadge reducibility of k-partitions. Journal of Logic and Algebraic Programming 79 (1) 92102.

V. Selivanov (2013) Total representations. Logical Methods in Computer Science 9 (2) 130.

S. Solecki (1998) Decomposing Borel sets and functions and the structure of Baire class 1 functions. Journal of the American Mathematical Society 11 (3) 521550.

F. van Engelen , A. W. Miller and J. Steel (1987) Rigid Borel sets and better quasi-order theory. In: Logic and Combinatorics (Arcata, California, 1985), Contemporary Mathematics 65 199222. (American Mathematical Society, Providence, RI.)

K. Weihrauch (2000) Computable analysis, An introduction. Texts in Theoretical Computer Science. An EATCS Series, Springer, Berlin.

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: 7 *
Loading metrics...

Abstract views

Total abstract views: 55 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th March 2017. This data will be updated every 24 hours.