Spectral Spaces and Ultrafilters

Abstract

In memory of my father. Let X be the prime spectrum of a ring. In Fontana and Loper [5 Fontana , M. , Loper , K. A. ( 2008 ). The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring . Comm. Algebra 36 : 2917 – 2922 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] the authors define a topology on X by using ultrafilters and show that this topology is precisely the constructible topology. In this paper we generalize the construction given in Fontana and Loper [5 Fontana , M. , Loper , K. A. ( 2008 ). The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring . Comm. Algebra 36 : 2917 – 2922 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and, starting from a set X and a collection of subsets ℱ of X, we define by using ultrafilters a topology on X in which ℱ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several examples of spectral spaces.