Stably continuous semilattices in closure spaces

In this paper, we introduce the concept of S-closure spaces and demonstrate that they precisely generate stably continuous semilattices. Additionally, we define the notion of S-morphisms between S-closure spaces to represent Scott continuous functions between stably continuous semilattices. These developments establish an equivalence between the category of stably continuous semilattices and the category of S-closure spaces with S-morphisms as the morphisms. This result provides a method for representing stably continuous semilattices through the framework of closure spaces.