### On topological properties of ultraproducts of finite sets

by Sagi and Shelah. [SaSh:841]

Math Logic Quarterly, 2005

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.

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