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Quasi-metric spaces and point-free geometry. (English) Zbl 1112.54012

The authors propose a ‘point-free’ approach to metric spaces which differs substantially from what is usually understood by that term (that is, the approach via frames, in which the primitive entities are taken to be arbitrary open regions of the space, equipped with a notion of ‘diameter’). Here the guiding principle is that the primitive entities are to be thought of as nonempty bounded closed regions, and the structure with which they are equipped (a quasi-metric) mimics the Hausdorff quasi-metric (the ‘one-sided’ version of the classical Hausdorff metric) on such sets in a classical metric space. ‘Diameter’ becomes a derived concept, albeit an important one, and ‘points’ are realized as equivalence classes of ‘Cauchy sequences’ of regions. (Incidentally, one might query the use of the term ‘point-free’, since one of the authors’ axioms asserts that at least one such point exists; however, it is not clear what use they make of this axiom.)
The work seems to be at an early stage: in the present paper, the authors do not study any notion of morphism between the spaces they consider, nor do they discuss any actual geometrical applications.

MSC:

54E99 Topological spaces with richer structures
51F99 Metric geometry
54B20 Hyperspaces in general topology
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