Basic Concepts of Quantale-Enriched Topologies

Abstract

The categorical framework for our axioms of quantale-enriched topologies is the theory of modules in the monoidal category $$\textsf {Sup}$$ and its free right modules generated by power sets. To express the intersection axiom we introduce the structure of a quasi-magma on a quantale. By selecting appropriate quantales and their corresponding quasi-magmas, we show that some well-established mathematical structures become quantale-enriched topologies. These include, among others, the closed left ideal lattices of non-commutative $$C^*$$ algebras, lower regular function frames of approach spaces as well as quantale-valued topological spaces.