A topological completeness theorem for transfinite provability logic

We prove a topological completeness theorem for the modal logic \(\textsf{GLP}\) containing operators \(\{\langle \xi \rangle :\xi \in \textsf{Ord}\}\) intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence \(\phi \) consistent with \(\textsf{GLP}\) can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in \(\phi \).