We will review recent work by Beklemishev and Gabelaia concerning the topological semantics of GLP. GLP is a polymodal logic where the modal operators correspond to a sequence of provability predicates, increasing in strength. A surprising connection between provability and topology is given by an interpretation of GLP over the class of "scattered" spaces, where every subset contains an isolated point. This interpretation has some advantages over Kripke semantics; in particular, GLP is incomplete for the class of frames over which it is sound, but Beklemishev and Gabelaia have shown that it is indeed complete for its topological interpretations. In this talk we shall outline the techniques they use for proving this result.