ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

Hypertopologies and quasi-metrics

Jesús Rodríguez-López, Salvador Romaguera Universidad Politécnica de Valencia, Spain

We study the relation between the Wijsman topology of a quasi-pseudo-metric space and others like the Vietoris, proximal and Hausdorff quasi-uniform topology, respectively. In particular, we prove that:

if (X, T ) is a quasi-pseudo-metrizable topological space then its Vietoris topology is the supremum of all Wijsman topologies generated by all quasi-pseudo-metrics compatible with T .

Furthermore, we show that the Wijsman topology of a quasi-pseudo-metric space (X, d) coincides with the Hausdorff quasi-uniform topology if and only if the conjugate quasi-pseudo-metric d^{−1} is hereditarily precompact.