  Motivated by the model theory of higher order logics, a certain kind of
topological spaces had been introduced on ultraproducts. These spaces are
called ultratopologies. Ultratopologies provide a natural extra topological
structure for ultraproducts and using this extra structure some preservation
and characterization theorems had been obtained for higher order logics. The
purely topological properties of ultratopologies seem interesting on their own
right. Here we present the solutions of two problems of Gerlits and Sagi. More
concretely we show that
  (1) there are sequences of finite sets of pairwise different cardinality such
that in their certain ultraproducts there are homeomorphic ultratopologies and
  (2) one can always find a dense set in an ultratopology whose cardinality is
strictly smaller than the cardinality of the ultraproduct, provided that the
factors of the corresponding ultraproduct are finite.
