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The Isbell-hull of a di-space. (English) Zbl 1245.54023

Recall that a pair $\left(X,d\right)$ is called a ${T}_{0}$-quasi-metric space if $X$ is a set and $d:X×X\to \left[0,\infty \right)$ is a mapping satisfying

(i) $d\left(x,x\right)=0$ for each $x\in X$;

(ii) $d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$ for any $x,y,z\in X$; and

(iii) $d\left(x,y\right)=0=d\left(y,x\right)$ implies that $x=y$.

A ${T}_{0}$-quasi-metric space $M$ is said to be injective if it has the property that whenever $X$ is a ${T}_{0}$-quasi-metric space, $A$ is a subspace of $X$, and $f:A\to M$ is a nonexpansive map, then $f$ can be extended to a nonexpansive map $g:X\to M$. It is shown that, for every ${T}_{0}$-quasi-metric space $X$, there exists a ${T}_{0}$-quasimetric space $I\left(X\right)$ with the following properties:

(a) $I\left(X\right)$ is injective:

(b) $X$ is isometric to a subspace of $I\left(X$); and

(c) $I\left(X\right)$ is minimal with respect to (a) and (b), i.e., whenever $M$ is an injective ${T}_{0}$-quasi-metric space containing an isometric copy of $X$, then $M$ contains an isometric copy of $I\left(X\right)$.

The construction of $I\left(X\right)$ is similar to J. R. Isbell’s construction of the injective hull of a metric space $X$ in [Comment. Alath. Helv. 39, 65–76 (1964; Zbl 0151.30205)] (which is also known as the tight span of $X$ or as the hyperconvex hull of $X$).

##### MSC:
 54D35 Extensions of topological spaces (compactifications, supercompactifications, completions, etc.) 54E35 Metric spaces, metrizability 54E50 Complete metric spaces 54C15 Retractions of topological spaces 54E55 Bitopologies