  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.
