A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). We extend it here to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g. several Hausdorff-Kuratowski-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

[1]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[2]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[3]  William W. Wadge,et al.  Degrees of complexity of subsets of the baire space , 1972 .

[4]  Robert van Wesep,et al.  Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II: Wadge degrees and descriptive set theory , 1978 .

[5]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[6]  John R. Steel,et al.  Determinateness and the separation property , 1981, Journal of Symbolic Logic.

[7]  William W. Wadge,et al.  Reducibility and Determinateness on the Baire Space , 1982 .

[8]  A. Louveau,et al.  Some results in the wadge hierarchy of borel sets , 1983 .

[9]  Arnold W. Miller,et al.  Rigid Borel sets and better quasi-order theory , 1985 .

[10]  A. Kechris.  Classical descriptive set theory , 1987 .

[11]  Peter Hertling,et al.  Unstetigkeitsgrade von Funktionen in der effektiven Analysis , 1996 .

[12]  Jacques Duparc,et al.  Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank , 2001, Journal of Symbolic Logic.

[13]  Peter Hertling,et al.  Topological properties of real number representations , 2002, Theor. Comput. Sci..

[14]  Victor L. Selivanov,et al.  Wadge degrees of ω-languages of deterministic Turing machines , 2003 .

[15]  Vasco Brattka.  Effective Borel measurability and reducibility of functions , 2005, Math. Log. Q..

[16]  Klaus Weihrauch,et al.  Computable Analysis , 2005, CiE.

[17]  Victor L. Selivanov,et al.  Towards a descriptive set theory for domain-like structures , 2006, Theor. Comput. Sci..

[18]  Victor L. Selivanov,et al.  The quotient algebra of labeled forests modulo h-equivalence , 2007 .

[19]  Jean Saint Raymond.  Preservation of the Borel class under countable-compact-covering mappings , 2007 .

[20]  Victor L. Selivanov,et al.  Hierarchies of Δ02‐measurable k ‐partitions , 2007, Math. Log. Q..

[21]  Victor L. Selivanov,et al.  Fine hierarchies and m-reducibilities in theoretical computer science , 2008, Theor. Comput. Sci..

[22]  Victor L. Selivanov,et al.  A Gandy Theorem for Abstract Structures and Applications to First-Order Definability , 2009, CiE.

[23]  Victor L. Selivanov,et al.  Definability in the h-quasiorder of labeled forests , 2009, Ann. Pure Appl. Log..

[24]  Daisuke Ikegami.  Games in set theory and logic , 2010 .

[25]  Benedikt Löwe,et al.  Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II , 2011 .

[26]  Victor L. Selivanov,et al.  Fine hierarchies via Priestley duality , 2012, Ann. Pure Appl. Log..

[27]  Benedikt Löwe,et al.  Wadge degrees and projective ordinals , 2012 .

[28]  Philipp Schlicht,et al.  Continuous reducibility for the real line , 2012 .

[29]  Victor L. Selivanov,et al.  Total Representations , 2013, Log. Methods Comput. Sci..

[30]  Verónica Becher,et al.  Wadge hardness in Scott spaces and its effectivization , 2015, Math. Struct. Comput. Sci..

[31]  Philipp Schlicht,et al.  Wadge-like reducibilities on arbitrary quasi-Polish spaces , 2012, Mathematical Structures in Computer Science.

[32]  Yann Pequignot,et al.  A Wadge hierarchy for second countable spaces , 2015, Arch. Math. Log..

[33]  Victor L. Selivanov.  Towards the Effective Descriptive Set Theory , 2015, CiE.

[34]  Victor L. Selivanov,et al.  Towards a descriptive theory of cb0-spaces , 2014, Mathematical Structures in Computer Science.

[35]  Victor L. Selivanov.  Extending Wadge Theory to k-Partitions , 2017, CiE.

[36]  Takayuki Kihara,et al.  On the structure of the Wadge degrees of bqo-valued Borel functions , 2019, Transactions of the American Mathematical Society.

[37]  Mathieu Hoyrup,et al.  Computability on Quasi-Polish Spaces , 2019, DCFS.

[38]  Takayuki Kihara,et al.  Wadge-like degrees of Borel bqo-valued functions , 2019, ArXiv.

[39]  Mathieu Hoyrup,et al.  Descriptive Complexity on Non-Polish Spaces , 2020, STACS.

[40]  Victor Selivanov.  Q-Wadge degrees as free structures , 2020, Comput..

[41]  Jacques Duparc,et al.  THE WADGE ORDER ON THE SCOTT DOMAIN IS NOT A WELL-QUASI-ORDER , 2019, The Journal of Symbolic Logic.

[42]  Victor Selivanov.  Effective Wadge Hierarchy in Computable Quasi-Polish Spaces , 2021, Sibirskie Elektronnye Matematicheskie Izvestiya.