  For any ordinal \Lambda, we can define a polymodal logic GLP(\Lambda), with a
modality [\xi] for each \xi<\Lambda. These represent provability predicates of
increasing strength. Although GLP(\Lambda) has no Kripke models, Ignatiev
showed that indeed one can construct a Kripke model of the variable-free
fragment with natural number modalities. Later, Icard defined a topological
model for the same fragment which is very closely related to Ignatiev's.
  In this paper we show how to extend these constructions for arbitrary
\Lambda. More generally, for each \Theta,\Lambda we build a Kripke model
I(\Theta,\Lambda) and a topological model T(\Theta,\Lambda), and show that the
closed fragment of GLP(\Lambda) is sound for both of these structures, as well
as complete, provided \Theta is large enough.
