Bitopological and topological ordered k-spaces

Abstract

Domain theory, in theoretical computer science, needs to be able to handle function spaces easily. It also requires asymmetric spaces, and these are necessarily not T 1. At the same time, techniques used with the higher separation axioms are useful there (see [Topology Appl. 199 (2002) 241]). In order to handle all these requirements, we develop a theory of k-bispaces using bitopological spaces, which results in a Cartesian closed category. The other well-known way to combine asymmetry and separation is ordered topological spaces [Nachbin, Topology and Order, Van Nostrand, 1965]; we define the category of ordered k-spaces, which is isomorphic to that found among bitopological spaces.