Normality, Regularity and Contractive Realvalued Maps

For approach spaces normality has been studied from different angles. One way of dealing with it is by focussing on separation by realvalued contractive maps or, equivalently, on Katětov–Tong’s insertion. We call this notion approach normality. Another point of view is using the isomorphism between the category App of approach spaces and contractions and the category [InlineEquation not available: see fulltext.] of lax algebras for the ultrafilter monad and the quantale P+ and applying the monoidal definition of normality. We call this notion monoidal normality. Although both normality properties coincide for topological approach spaces, a comparison of both notions for App is an open question. In this paper we present a partial solution to this problem. We show that in App approach normality implies monoidal normality and that both notions coincide on the subcategory of quasimetric approach spaces. Moreover we investigate the relation between approach normality and regularity. Among other things we prove that approach spaces that are approach normal and regular are uniform.