October 8
Brent Cody, Virginia Commonwealth University
Higher derived topologies

By beginning with the order topology on an ordinal $\delta$, and iteratively declaring more and more derived sets to be open, Bagaria defined the derived topologies $\tau_\xi$ on $\delta$, where $\xi$ is an ordinal. He showed that the non-isolated points in the space $(\delta,\tau_\xi)$ can be characterized using a strong form of iterated simultaneous stationary reflection called $\xi$-s-reflection, which is deeply connected to certain transfinite indescribability properties. However, Bagaria's definitions break for $\xi\geq\delta$ because, under his definitions, the $\delta$-th derived topology $\tau_\delta$ is discrete and no ordinal $\alpha$ can be $\alpha+1$-s-stationary. We will discuss some new work in which we use certain diagonal versions of Bagaria's definitions to extend his results. For example, we introduce the notions of diagonal Cantor derivative and use it to obtain a sequence of derived topologies on a regular $\delta$ that is strictly longer than that of Bagaria's, under certain hypotheses.