Continuity on directed spaces

Abstract

We first investigate two approximation relations on a T 0 topological space, the n-approximation and the d-approximation, which are generalizations of the waybelow relation on a dcpo. Different kinds of continuous spaces are defined by the two approximations and are all shown to be directed spaces. We will show that the continuity of a directed space is very similar to the continuity of a dcpo in many aspects, which indicates that the notion of directed spaces is a suitable topological extension of dcpos.The main results are: (1) A topological space is continuous iff it is a retract of an algebraic space.(2) a directed space X is core compact iff for any directed space Y , X × Y = X ⊗ Y , where X × Y and X ⊗ Y are the topological product and categorical product in DTop of X and Y respectively; (3) a directed space is continuous (resp., algebraic, quasicontinuous, quisialgebraic) iff the lattice of its closed subsets is continuous (resp., algebraic, quasicontinuous, quisialgebraic).

A note on the category of c-spaces

“…There are many important directed spaces in domain theory, for instance locally finitary compact spaces are directed spaces; in particular c-spaces and Alexandroff spaces are directed spaces. Since the category of continuous domains is not cartesian closed, and since the position of the category of c-spaces in the category of directed spaces is similar to that of continuous domains in dcpos [3], a natural question arises: Is the category of c-spaces cartesian closed? In this short note, we answer this question in the negative.…”