Filter descriptive classes of Borel functions
Volume 204 / 2009
Fundamenta Mathematicae 204 (2009), 189-213
MSC: Primary 03E15; Secondary 03E45, 54A20, 54H05.
DOI: 10.4064/fm204-3-1
Abstract
\We first prove that given any analytic filter ${\cal F}$ on $\omega$ the set of all functions $f$ on ${\bf 2}^\omega$ which can be represented as the pointwise limit relative to ${\cal F}$ of some sequence $ (f_{n})_{n\in\omega}$ of continuous functions ($f=\lim_{\cal F} f_n$), is exactly the set of all Borel functions of class $\xi$ for some countable ordinal $\xi$ that we call the rank of ${\cal F}$. We discuss several structural properties of this rank. For example, we prove that any free $\Pi^0_ 4$ filter is of rank 1.
