Semi-Lipschitz Functions, Best Approximation,
and Fuzzy Quasi-Metric Hyperspaces



José Manuel Sánchez Álvarez


Motivated, in part, by some problems from functional analysis, concentration
of measures, dynamical systems, theoretical computer science, mathematical
economics, etc, in the last years a mathematical theory has been developed
in order to generalize classical mathematical theories: hyperspaces, function
spaces, topological algebra, etc.


This doctoral thesis is devoted to study some of these generalizations from
a nonsymmetric point of view. In the ¯rst part, we study the set of semi-
Lipschitz functions, we show that this set can be endowed with the structure of
normed cone. We also study different types of completeness, (bicompleteness,
right k-completeness, D-completeness), and we explore when the corresponding
quasi-distance is balanced. Using relativized norms we present a model for
computing the complexity of certain algorithms, which is done with the help of
a suitable space of semi-Lipschitz functions. On the other hand, we show that
our approach provides an appropriate setting to characterize the points of best
approximation of quasi-metric spaces.


The fact that some hypertopologies have been successfully applied to sev-
eral areas of Computer Science has contributed to increase the interest of a
nonsymmetric study of hypertopologies. Thus, in a second part, we study some
conditions on best approximation in the realm of quasi-metric hyperspaces. By
other hand, we characterize completeness of a uniform space using Sieber-Pervin
completeness, Smyth completeness and D-completeness of its upper Hausdorff-
Bourbaki quasi-uniformity, on the collection of its nonempty compact subsets.


Finally we introduce two notions of fuzzy quasi-metric hyperspace that
generalize the corresponding notions of fuzzy metric space by Kramosil and
Michalek, and by George and Veeramani respectively, to the quasi-metric hy-
perspace context. Several basic properties of completeness, precompactness,
and compactness of these spaces are obtained. We apply this theory to some
examples and we point out some advantages of the use of fuzzy quasi-metrics
instead of classical metrics and quasi-metrics.



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