Separation Classes of Equilogical Spaces

The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category ASSM of assemblies over algebraic lattices, which is a generalisation of Scott's category EQU of equilogical spaces. In this paper, we identify cartesian closed subcategories of assemblies which correspond to well-known separation properties of topology: T 0 , T 1 , Hausdorff, completely Hausdorff, totally disconnected, completely regular, zero-dimensional. 1 Introduction It is well-known that the categories TOP of topological spaces and TOP 0 of T 0 topological spaces are not cartesian closed, i.e., do not admit a function space construction such that typed -calculus could be interpreted in the category. In December 1996, Dana Scott proposed to remedy the situation by extending TOP 0 to the cartesian closed category EQU of equilogical spaces [12, 1, 4]. There are several equivalent descriptions of EQU; one of them is by modest sets over algebraic lattices [1]...