On quasi-metrizable d-spaces

Abstract

In this paper we develop some connections between the quasi-metric spaces and the d-spaces of domain theory. We do this by introducing the notion of S-quasi-metrics on dcpos, which is a quasi-metric compatible with the ordering. Results show that (i) the quasi-metrizable d-spaces are exactly the S-quasi-metric spaces; (ii) the open ball topology is generally coarser than the Scott topology in an S-quasi-metric space, and a condition is provided to make them coincide; (iii) the Scott space of formal balls of a complete metric space is S-quasi-metrizable; (iv) the quasi-metrizability, as a topological property, is generally not preserved by the D-completion, the well-filtered reflection, or the sobrification of a T0 space.