Wadge degrees and descriptive set theory

Wesep, Edward Dickersin Van

doi:10.1007/bfb0069298

Cited by 66 publications

(37 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Kechris has proved in ZF + DC that a class T and its dual f cannot both have uniform separation, provided that T is closed under 3W, VR, and continuous preimages. This gives a new proof of a weak form of Van Wesep's theorem [8] that AD implies T and T cannot both have separation. Harrington has proved that if ZFC is consistent then it is consistent with ZFC that both £í¡ and II3 have separation.…”

Section: Theoremmentioning

confidence: 87%

A technique for proving uniformity

Becker¹

1984

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract.We introduce a general technique for showing that if various properties hold, then they hold uniformly, assuming some determinacy. Uniform means in the boldface sense, that is, there are continuous functions taking indices to indices. For example, if T is a parametrized pointclass then all closure properties of A are uniform, and the separation property for T implies uniform separation.Our notation and terminology is that of Moschovakis [5], Let T be an Rparametrized (boldface) pointclass, closed under continuous preimages. Any pointclass which has a parametrization (that is, a universal set) also has a good parameterization, i.e. one for which there are continuous S-m-n functions [5, 3H

show abstract

Section: Theoremmentioning

confidence: 87%

A technique for proving uniformity

Becker¹

1984

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…He has also computed the rank ν of this structure which is a rather large ordinal. In [VW76,St80] the following deep relation of the Wadge reducibility to the separation property was established: For any Borel set A which is non-self-dual (i.e., A ≤ W A) exactly one of the principal ideals {X | X ≤ W A}, {X | X ≤ W A} has the separation property. The mentioned results give rise to the Wadge hierarchy which is, by definition, the sequence {Σ α } α<ν of all non-self-dual principal ideals of (∆ 1 1 (N ); ≤ W ) that do not have the separation property and satisfy for all α < β < ν the strict inclusion Σ α ⊂ ∆ β .…”

Section: Fh Of K-partitions and Wadge-like Reducibilitiesmentioning

confidence: 99%

Towards a descriptive theory of cb₀-spaces

Selivanov

2016

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

The paper tries to extend results of the classical Descriptive Set Theory to as many countably based T0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we consider also the more general case of k-partitions. In particular, we investigate the difference hierarchy of k-partitions and the fine hierarchy closely related to the Wadge hierarchy.

show abstract

“…Wadge also computed the corresponding (large) ordinal ν. In [4,5], it was shown that for any non self-dual Borel-Wadge class C, exactly one of the classes C or co(C) has the separation property (see [6]). Here, co(C) = {X | X ∈ C} is a dual class for C.…”

Section: Preliminariesmentioning

confidence: 99%

Classifying countable Boolean terms

Selivanov

2005

Algebr Logic

View full text Add to dashboard Cite

We deal with the Borel and difference hierarchies in the space P ω of all subsets of ω endowed with the Scott topology. (The spaces P ω and 2 ω coincide set-theoretically but differ topologically.) We look at the Wadge reducibility in P ω. The results obtained are applied to the problem of characterizing ω 1 -terms t which satisfy C = t(Σ 0 1 ) for a given Borel-Wadge class C. We give its solution for some levels of the Wadge hierarchy, in particular, all levels of the Hausdorff difference hierarchy. Finally, we come up with a discussion of some relevant facts and open questions. PRELIMINARIESLet {Σ 0 α } α<ω1 be the Borel hierarchy of subsets in the Cantor space 2 ω ; here, ω 1 is the first noncountable ordinal. As usual, Π 0 α is a dual class for Σ 0 α and ∆ 0 α = Σ 0 α ∩ Π 0 α is the corresponding self-dual class. The class of all Borel sets is denoted by B = α<ω1 Σ 0 α . Levels of the Borel hierarchy, as well as many other classes studied within descriptive set theory, may be defined by means of suitable -generally infinitary -set-theoretic operations. We define a natural class of such operations, which we call ω 1 -terms, by induction as follows: constants 0, 1 and variables v k (k < ω) are ω 1 -terms; if t i (i < ω) are ω 1 -terms, then so are the expressionst 0 , t 0 ∪ t 1 , t 0 ∩ t 1 , i<ω t i , and i<ω t i .If t = t(v k ) is an ω 1 -term, then t({A k }) denotes the value of t whenever each variable v k (k < ω) is interpreted as some set A k ⊆ 2 ω . Let t(Σ 0 1 ) be the set of all values t({A k }), where A k ∈ Σ 0 1 for any k < ω. Similarly, we write t(C) for other kinds of terms t and classes C of sets.It turns out that classes like t(Σ 0 1 ) admit of a very natural description in terms of Wadge reducibility, which is the preorder ≤ W on the class P (2 ω ) of all subsets of 2 ω defined as follows:for some continuous function f : 2 ω → 2 ω . By a Wadge class we mean any principal ideal of the form {X | X ≤ W A}, for a given A ⊆ 2 ω . Such a class is said to be Borel if A is Borel, and we refer to it as nonTHEOREM 1.1 [1]. Any C ⊆ P (2 ω ) is a non self-dual Borel-Wadge class if and only if C = t(Σ 0 1 ) for some ω 1 -term t.The theorem follows from a relevant result in [2,3].

show abstract

Wadge degrees and descriptive set theory

Cited by 66 publications

References 2 publications

A technique for proving uniformity

A technique for proving uniformity

Towards a descriptive theory of cb₀-spaces

Classifying countable Boolean terms

Contact Info

Product

Resources

About

Wadge degrees and descriptive set theory

Cited by 66 publications

References 2 publications

A technique for proving uniformity

A technique for proving uniformity

Towards a descriptive theory of cb0-spaces

Classifying countable Boolean terms

Contact Info

Product

Resources

About

Towards a descriptive theory of cb₀-spaces