Higher stationarity and derived topologies on ${\mathcal{P}}_{\kappa }(A)$

Let $\kappa$ be a regular limit cardinal, $\kappa \subseteq A$. We study a notion of $n$-s-stationarity on ${\mathcal{P}}_{\kappa }(A)$. We construct a sequence of topologies $⟨{\tau }_0,{\tau }_1,\dots ⟩$ on ${\mathcal{P}}_{\kappa }(A)$ characterising the simultaneous reflection of a pair of $n$-s-stationary sets in terms of elements in the base of ${\tau }_n$. This result constitutes a complete generalisation to the context of ${\mathcal{P}}_{\kappa }(A)$ of Bagaria’s prior characterisation of $n$-simultaneous reflection in terms of derived topologies on ordinals.