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

Recall that a pair (X,d) is called a T 0 -quasi-metric space if X is a set and d:X×X[0,) is a mapping satisfying

(i) d(x,x)=0 for each xX;

(ii) d(x,z)d(x,y)+d(y,z) for any x,y,zX; and

(iii) d(x,y)=0=d(y,x) 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:AM is a nonexpansive map, then f can be extended to a nonexpansive map g:XM. It is shown that, for every T 0 -quasi-metric space X, there exists a T 0 -quasimetric space I(X) with the following properties:

(a) I(X) is injective:

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

(c) I(X) 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(X).

The construction of I(X) 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:
54D35Extensions of topological spaces (compactifications, supercompactifications, completions, etc.)
54E35Metric spaces, metrizability
54E50Complete metric spaces
54C15Retractions of topological spaces
54E55Bitopologies
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