Spectral Spaces and Ultrafilters

In memory of my father. Let X be the prime spectrum of a ring. In Fontana and Loper [5] 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] 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. © 2014 Copyright Taylor and Francis Group, LLC.