Covering dimension and finite-to-one maps

Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most 2n-to-1 map

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