Hyperspaces of a weightable quasi-metric space: Application to models in the theory of computation

It is well known that both weightable quasi-metrics and the Hausdorff distance provide efficient tools in several areas of Computer Science. This fact suggests, in a natural way, the problem of when the upper and lower Hausdorff quasi-pseudo-metrics of a weightable quasi-metric space ( X , d ) are weightable. Here we discuss this problem. Although the answer is negative in general, we show, however, that it is positive for several nice classes of (nonempty) subsets of X . Since the construction of these classes depends, to a large degree, on the specialization order of the quasi-metric d , we are able to apply our results to some distinguished quasi-metric models that appear in theoretical computer science and information theory, like the domain of words, the interval domain and the complexity space.