On equivalent category bases
John Morgan · Pacific Journal of Mathematics · 1983
Any category base in which every region contains a minimal region is equivalent to a topology.Utilizing the notion of a category base, which is a generalization of the notion of a topology, the author has developed a general theory of point sets within which many of the analogies between Lebesgue measure and Baire category have been unified (cf.[3]-[9]).In view of the "equivalence" between the Lebesgue measurable sets and the sets having the Baire property relative to the density topology (cf.[10, Chapter 22]), the question arises whether every category base is equivalent to some topology.We do not know the answer to this question.However, we shall show in this article that for a certain class of category bases, including all finite category bases, there do exist equivalent topologies.After stating pertinent facts in §1, we define in §2 a basic topology which is associated with a given category base.In §3 we determine this basic topology in several examples and see that it is not in general equivalent to the category base from which it arises.As we show in §4, however, any category base in which every region contains a minimal region is equivalent to its basic topology.