DS-partial metric spaces and domain theory

Abstract

In this paper, we establish some connections between partial metric spaces and domain theory, and give a characterization of stable partially metrizable d-spaces. First, the concept of S-partial metrics on posets is introduced. Then it is demonstrated that the open ball topology is coarser than the Scott topology in an S-partial metric space, and that the partially metrizable d-space is exactly the DS-partial metric space. In addition, some conditions are provided to ensure that the open-ball topology and the Scott topology coincide in an S-partial metric space. Moreover, we prove that, for an S-partial metric space (X,≤,p), (X,Op(X)) is sober iff the set of all limit points of every self-convergent sequence is directed. Finally, we propose the notion of DS-valuation space and obtain the result that there is a bijection between stable DS-partial metric spaces and DS-valuation spaces, thereby giving the characterization of stable partially metrizable d-spaces.