wadge reducibility
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2019 ◽  
Vol 54 ◽  
pp. 45-63
Author(s):  
Riccardo Camerlo
Keyword(s):  

It is proved that the Tang-Pequignot reducibility (or reducibility by relatively continuous relations) on a second countable, T0 space X either coincides with the Wadge reducibility for the given topology, or there is no topology on X that can turn it into Wadge reducibility.



2010 ◽  
Vol 79 (1) ◽  
pp. 92-102 ◽  
Author(s):  
Victor L. Selivanov
Keyword(s):  


2008 ◽  
Vol 2 (1) ◽  
pp. 5-36 ◽  
Author(s):  
Victor Selivanov
Keyword(s):  


2008 ◽  
Vol 19 (03) ◽  
pp. 649-675 ◽  
Author(s):  
VICTOR SELIVANOV

We develop a theory of regular aperiodic ω-languages in parallel with the theory around the Wagner hierarchy. In particular, we characterize the Wadge degrees of regular aperiodic ω-languages, find an effective version of the Wadge reducibility adequate for this class of languages and prove "aperiodic analogs" of the Büchi-Landweber determinacy theorem and of the Landweber's characterization of regular open and regular Gδ sets.



2008 ◽  
Vol 202 ◽  
pp. 59-71 ◽  
Author(s):  
Victor Selivanov
Keyword(s):  


2005 ◽  
Vol 120 ◽  
pp. 159-171 ◽  
Author(s):  
Victor L. Selivanov
Keyword(s):  


1987 ◽  
Vol 52 (1) ◽  
pp. 232-242
Author(s):  
Randall Dougherty

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.



1984 ◽  
Vol 49 (3) ◽  
pp. 774-782 ◽  
Author(s):  
Douglas Cenzer

AbstractLet A and B be subsets of the space 2N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2N into 2N such that A = Φ−1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, a ⊂ b implies Φ(a) ⊂ Φ(b). The set A is said to be monotone if a ∈ A and a ⊂ b imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The sets are all reducible to the ( but not ) sets, which are in turn all reducible to the strictly sets, which are all in turn reducible to the strictly sets. In addition, the nontrivial sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly monotone sets which have different monotone degrees. We show that every monotone set is actually positive. We also consider reducibility for subsets of the space of compact subsets of 2N. This leads to the result that the finitely iterated Cantor-Bendixson derivative Dn is a Borel map of class exactly 2n, which answers a question of Kuratowski.



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