On topological properties of ultraproducts of finite sets
Gábor Sági, Saharon Shelah · Mathematical logic quarterly · 2005
In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. Here we present the solutions of two such problems. More concretely we show 1. that 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|. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)