Approaching metric domains

In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T${}_0$ spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T${}_0$ approach spaces and characterise them as precisely the continuous lattices equipped with an unitary and associative $[0,\mathrm{\infty }]$-action. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces.