Notes on Smyth-completes and local Yoneda-completes

In this paper, we first introduce the notion of d-net which is obtained being inspired by an elegant characterization: a quasi-metric space $(X,d)$ is Smyth-complete if and only if $B(X,d)$ is sober in its open ball topology. Then, we obtain the characterizations of Smyth-complete quasi-metric spaces via d-nets in quasi-metric spaces. Or rather, we prove that a quasi-metric space is Smyth-complete if and only if every d-net has a d-limit and converges to its d-limit. For a local Yoneda-complete quasi-metric space, we provide a necessary and sufficient condition such that the open ball topology coincides with the Scott topology on its formal ball space. In addition, we show that local Yoneda-completeness is preserved by some constructions, such as coproducts, products, function spaces, formal ball spaces and so on. Finally, we prove that the formal ball construction B induces the monads on the categories of Smyth-complete quasi-metric spaces with 1-Lipschitz maps, local Yoneda-complete quasi-metric spaces with 1-Lipschitz maps, Smyth-complete quasi-metric spaces with Y-continuous maps and local Yoneda-complete quasi-metric spaces with Y-continuous maps.