# 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…

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## References

SHOWING 1-10 OF 62 REFERENCES

### Continuous reducibility for the real line

- Mathematics
- 2012

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

- MathematicsThe Journal of Symbolic Logic
- 2013

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.

### Decomposing Borel functions and structure at finite levels of the Baire hierarchy

- MathematicsAnn. Pure Appl. Log.
- 2012

### Weihrauch degrees, omniscience principles and weak computability

- Mathematics, Computer ScienceThe Journal of Symbolic Logic
- 2011

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

- Mathematics
- 1998

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

- MathematicsElectron. Notes Theor. Comput. Sci.
- 2008

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

- MathematicsCiE
- 2009

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.