Set-Valued Analysis

, Volume 11, Issue 4, pp 323–344 | Cite as

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

  • Jesús Rodríguez-López
  • Salvador Romaguera


The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P0(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P0(X) if and only if d−1 is hereditarily precompact.

quasi-pseudo-metric (proximal) hit-and-miss topologies Wijsman topology Vietoris topology far Urysohn family uniformly Urysohn family Hausdorff quasi-pseudo-metric 


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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Jesús Rodríguez-López
    • 1
  • Salvador Romaguera
    • 1
  1. 1.Escuela Politécnica Superior de Alcoy, Departamento de Matemática AplicadaUniversidad Politécnica de ValenciaAlcoy (Alicante)Spain

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