On topological properties of ultraproducts of finite sets

Motivated by the model theory of higher order logics, in [2] 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 in [2] some preservation and characterization theorems had been obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [3], where some questions remained open. Here we present the solutions of two such problems. More concretely we show that (1) there are sequences of finite sets of pairwise different cardinalities such that in their certain ultraproducts there are homeomorphic ultratopologies and (2) if A is an infinite ultraproduct of finite sets then every ultratopology on A contains a dense subset D such that |D| < |A|. AMS Classification: 03C20, 54A25, 54A99.