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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 $[0,\infty]$-action. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces.