Derived Topologies on Ordinals and Stationary Reflection

We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor’s derivative operator on sets of ordinals, starting form the usual interval topology. We characterize the non-isolated points in the ξ-th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call ξ-simultaneous-reflection. We prove some properties of the ideals of non-ξ-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of Πξ-indescribability, for any ordinal ξ, which extends to the transfinite the usual notion of Πn-indescribability, and prove that in the constructible universe L, a regular cardinal is (ξ + 1)-simultaneouslyreflecting if and only if it is Πξ-indescribable, a result that generalizes to all ordinals ξ previous results of Jensen [28] in the case ξ = 2, and Bagaria-Magidor-Sakai [5] in the case ξ = n. This yields a complete characterization in L of the non-discreteness of the ξ-topologies, both in terms of iterated stationary reflection and in terms of indescribability.