  Provability logics are modal or polymodal systems designed for modeling the
behavior of G\"odel's provability predicate in arithmetical theories and its
natural extensions. If \Lambda is any ordinal, the G\"odel-L\"ob calculus
GLP(\Lambda) contains one modality [\lambda] for each \lambda<\Lambda,
representing provability predicates of increasing strength. GLP(\Lambda) has no
Kripke models, but Beklemishev and Gabelaia recently proved that GLP(\omega) is
complete for its class of topological models.
  In this paper we generalize Beklemishev and Gabelaia's result to GLP(\Lambda)
for arbitrary \Lambda. We also introduce provability ambiances, which are
topological models where valuations of formulas are restricted. With this we
show completeness of GLP(\Lambda) for the class of provability ambiances based
on Icard polytopologies.
