Scott approach distance on metric spaces

The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are investigated, including its topological coreflection and its relation to injective $T_0$ approach spaces. It is proved that the topological coreflection of the Scott distance is sandwiched between the $d$-Scott topology and the generalized Scott topology; and that every injective $T_0$ approach space is a cocomplete and continuous metric space equipped with its Scott distance.