Recall that a pair is called a -quasi-metric space if is a set and is a mapping satisfying
(i) for each ;
(ii) for any ; and
(iii) implies that .
A -quasi-metric space is said to be injective if it has the property that whenever is a -quasi-metric space, is a subspace of , and is a nonexpansive map, then can be extended to a nonexpansive map . It is shown that, for every -quasi-metric space , there exists a -quasimetric space with the following properties:
(a) is injective:
(b) is isometric to a subspace of ); and
(c) is minimal with respect to (a) and (b), i.e., whenever is an injective -quasi-metric space containing an isometric copy of , then contains an isometric copy of .
The construction of is similar to J. R. Isbell’s construction of the injective hull of a metric space in [Comment. Alath. Helv. 39, 65–76 (1964; Zbl 0151.30205)] (which is also known as the tight span of or as the hyperconvex hull of ).