April 25
Catalina Torres Pachon,
University of Barcelona
A Topological Approach to Characterising Hyperstationary Sets on $\mathcal{P}_\kappa(A)$
Given a topological space $(X, \tau)$, the derived set operator $d_\tau$ maps a set $T$ to its set of limit points with respect to $\tau$. Fixing an initial topology $\tau_0$ on $X$, we can define a sequence of derived topologies $(X, \tau_0, \tau_1, \dots, \tau_\xi, \dots)$, where $\tau_\zeta \subseteq \tau_\xi$ for $\zeta \lt \xi$. This is achieved by declaring $d_{\tau_\xi}(T)$ to be open in $\tau_{\xi+1}$ and taking unions at limit stages.
In Derived Topologies on Ordinals and Stationary Reflection, Bagaria characterised the non-isolated points in the $\xi$-th derived topology on ordinals as those satisfying a strong iterated form of stationary reflection, termed $\xi$-simultaneous reflection.
Generalisations of combinatorial properties of ordinals to $\mathcal{P}_\kappa(A) := \{X \subseteq \kappa : |X| \lt \kappa\}$, where $\kappa$ is an uncountable regular cardinal and $A \subseteq \kappa$, have been widely studied. In this context, we extend the notion of higher stationarity and construct a sequence of topologies $\langle \tau_0, \tau_1, \dots \rangle$ on $\mathcal{P}_\kappa(A)$, characterising the simultaneous reflection of high-stationary subsets of $\mathcal{P}_\kappa(A)$ in terms of elements in the base of a derived topology on $\mathcal{P}_\kappa(A)$.