Effective Mathematics of the Uncountable, New York, August 2009

Abstract: It is natural to do descriptive set theory on the $\omega_1$-Baire space, that is, $(\omega_1)^{\omega_1}$, with the topology generated by the countably determined sets. Vaananen and others have done a great deal of work to prove analogs of classical theorems for this space.

In this space, the closest analog of the Borel sets are the $\omega_1$-Borel sets, that is, the smalest $\omega_2$-algebra contaning the open sets. Unfortunately, this space lacks a “separation” theorem, so that the $\omega_1$-Borel sets are a paltry subclass of the $\mathbf{\Delta}^1_1$ sets. There is a wide gap in between them, and we will present a number of open questions concerning this gap.