Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.Comment: 50 pages, revised version, accepted for publication on Mathematical Structures in Computer Scienc

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

Continuous reducibility and dimension of metric spaces

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,\tau)$ is called the \emph{Wadge quasi-order} for $(X,\tau)$. We further show that this quasi-order, restricted to the Borel subsets of a Polish space $(X,\tau)$, is a \emph{well-quasiorder (wqo)} if and only if $(X,\tau)$ has dimension $0$, as an application of the main result. Moreover, we give further examples of applications of the technique, which is based on a construction of graph colorings

Total Representations

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.Comment: 30 page

On the structure of finite level and \omega-decomposable Borel functions

We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of \Sigma^0_\alpha-measurable functions (for every fixed 1 \leq \alpha < \omega_1). Moreover, we present some results concerning those Borel functions which are \omega-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.Comment: 31 pages, 2 figures, revised version, accepted for publication on the Journal of Symbolic Logi