Hyperspaces, quasi-uniformities and quasi-metrics

Several useful properties used in Computer Science are quite different from those considered in classical mathematics. In this way, the spaces which are interesting in the applications to this science are defined in general from partial ordered objects which represent stages of a computational process. Consequently, the quasi-uniform and quasi-metric spaces are the suitable context to interpret several interesting properties of such processes.;The fact that some hypertopologies have been successfully applied to several areas of Computer Science has contributed to increase the interest of a nonsymmetric study (quasi-metrics, quasi-uniformities) of hypertopologies. However, several problems remained without satisfactory solution.;This doctoral thesis is devoted to study several hypertopologies from a nonsymmetric point of view. In the first part, we study several well-known hypertopologies as the Vietoris, proximal, Wijsman, Fell and Hausdorff quasi-uniform hypertopologies respectively. In this way, we prove that in a quasi-uniform space (X, U ) the Vietoris topology agrees with the Hausdorff quasi-uniform topology on the set K0(X) of all nonempty compact subsets of (X, TU ) if and only if U-1 |K is precompact for all K ∈ K0(X). As a consequence of our results, we obtain several facts for the symmetric case. In fact, we prove that in a metric space (X, d) the Vietoris topology coincides with the Hausdorff metric topology on P0 (X) if and only if X is finite.;The nonsymmetric study of the Fell topology motivates the introduction of the double topological spaces generalizing the notion of a bitopological space.;Furthermore, we study bitopological notions of the Fisher and Kuratowski-Painleve convergence respectively. We also obtain results about their relationships with the above hypertopologies.;In the second part, we study several hypertopologies and convergence of functions in the space of all continuous functions between two topological spaces. One of our main results proves that in a quasi-pseudo-metric space, the upper Hausdorff quasi-pseudo-metric topology agrees with the proximal topology if and only if every real-valued lower semicontinuous function is quasi-uniformly continuous.