We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.

We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions of (...) continuous functions with domains of type Πn0, for a fixed n<ω. For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n=1 and Semmes for n=2. (shrink)

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.