Finer topologies on pointsets in Polish spaces

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.