Wadge-like reducibilities on arbitrary quasi-Polish spaces

@article{Ros2014WadgelikeRO,
  title={Wadge-like reducibilities on arbitrary quasi-Polish spaces},
  author={Luca Motto Ros and Philipp Schlicht and Victor L. Selivanov},
  journal={Mathematical Structures in Computer Science},
  year={2014},
  volume={25},
  pages={1705 - 1754}
}
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 Δ0 α-reductions, and try to find for various natural topological spaces X the least ordinal α X such that for every α X ⩽ β < ω1 the degree-structure induced on X by the Δ0 β-reductions is simple (i.e… 

Bad Wadge-like reducibilities on the Baire space

We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently,

Better-quasi-order : ideals and spaces

This thesis deals with combinatorics, order theory and descriptive set theory. The first contribution is to the theory of well-quasi-orders (wqo) and better-quasi-orders (bqo). The main result is the

Continuous reducibility and dimension of metric spaces

TLDR
It is shown that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X) that are pairwise incomparable with respect to continuous reducibility.

The Wadge hierarchy on Zariski topologies

A Wadge hierarchy for second countable spaces

TLDR
A notion of reducibility for subsets of a second countable T0 topological space based on relatively continuous relations and admissible representations is defined and induces a hierarchy that refines the Baire classes and the Hausdorff–Kuratowski classes of differences.

On the Structure of Finite Level and ω-Decomposable Borel Functions

  • L. Ros
  • Mathematics
    The Journal of Symbolic Logic
  • 2013
TLDR
A full description of the structure under inclusion of all finite level Borel classes of functions is given, and an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σ α 0-measurable functions is provided.

THE WADGE ORDER ON THE SCOTT DOMAIN IS NOT A WELL-QUASI-ORDER

Abstract We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose,

Wadge-like degrees of Borel bqo-valued functions

TLDR
The main result states that the structure of the $\mathbf{\Delta}^0_\alpha$-degrees of $Q$-measurable $Q-valued functions is isomorphic to the generalized homomorphism order on the $\gamma$-th iterated £Q-labeled forests.

Borel subsets of the real line and continuous reducibility

We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the

Sailing Routes in the World of Computation

TLDR
The tutorial focuses on computably enumerable (c.e.) structures, a class that properly extends the class of all computable structures and the interplay between important constructions, concepts, and results in computability, universal algebra, and algebra.

References

SHOWING 1-10 OF 62 REFERENCES

Quasi-Polish spaces

Continuous reducibility for the real line

We study Borel subsets of the real line up to continuous reducibility. We firstly show that every quasi-order of size ω1 embeds into the quasiorder of Borel subsets of the real line up to continuous

On the Structure of Finite Level and ω-Decomposable Borel Functions

  • L. Ros
  • Mathematics
    The Journal of Symbolic Logic
  • 2013
TLDR
A full description of the structure under inclusion of all finite level Borel classes of functions is given, and an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σ α 0-measurable functions is provided.

Definability in the h-quasiorder of labeled forests

On the Wadge Reducibility of k-Partitions

  • V. Selivanov
  • Mathematics
    Electron. Notes Theor. Comput. Sci.
  • 2008

Weihrauch degrees, omniscience principles and weak computability

TLDR
It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn–Banach Theorem in this new and very strong sense and any single-valued weakly computable operation is already computable in the ordinary sense.

Decomposing Borel sets and functions and the structure of Baire class 1 functions

All spaces considered are metric separable and are denoted usually by the letters X, Y, or Z. w stands for the set of all natural numbers. If a metric separable space is additionally complete, we

On the Difference Hierarchy in Countably Based T0-Spaces

  • V. Selivanov
  • Mathematics
    Electron. Notes Theor. Comput. Sci.
  • 2008

A Gandy Theorem for Abstract Structures and Applications to First-Order Definability

TLDR
A Gandy theorem is established that for any k *** 3 a predicate on the quotient structure of the h -quasiorder of finite k -labeled forests is definable iff it is arithmetical and invariant under automorphisms.
...