Scott quasi-metric and Scott quasi-uniformity based on pointwise quasi-metrics

In this paper, we develop some connections between pointwise quasi-metric spaces and Scott spaces in domain theory. The main results include (i) the category of Scott quasi-metrics with S-morphisms is equivalent to that of pointwise quasi-metrics in the sense of Shi; (ii) a topological space $(X,T)$ is quasi-metrizable if and only if the topologically generated space $(I^X,{\omega }_I(T\left)\right)$ (where ${\omega }_I\left(T\right)$ denotes the family of all lower semi-continuous mappings from X to the unit interval I) can be induced by a pointwise quasi-metric with a property M; (iii) the notion of Scott quasi-uniformity is presented, and it is shown that d-spaces of domain theory are exactly the Scott quasi-uniformizable spaces; (iv) the relationship between Scott quasi-metrics (introduced by the first and second authors) and Scott quasi-uniformities is established. In specific, the Scott quasi-metrics are exactly the Scott quasi-uniformities that has a countable base.