Finer topologies on pointsets in Polish spaces
Volume 261 / 2023
Fundamenta Mathematicae 261 (2023), 99-131
MSC: Primary 03E15; Secondary 03E60.
DOI: 10.4064/fm211-11-2022
Published online: 30 January 2023
Abstract
Assuming $\mathrm{AD}_{\mathbb R}$, we generalize to higher levels some classical theorems about the first level of the projective hierarchy. The theorems involve the relationships between a pointclass and the type of finer topology that pointsets in that pointclass admit. One such classical theorem is this: A pointset $Y$ in a Polish space is Borel iff there exists a Polish topology on $Y$ finer than the usual topology.
