On metrization of the hit-or-miss topology using Alexandroff compactification

AbstractWhen the hit-or-miss topology is employed, the space of all closed subsets of a Hausdorff, locally compact and second countable space (HLCSC) is known to be Hausdorff, compact and second countable, thus metrizable. This paper investigates metrics on this space by using Alexandroff compactification technique, with a more general metrization procedure developed. A note concerning the necessity of the condition HLCSC on E is included. With the constructed metric, we investigate a hyperspace Birkhoff ergodic theorem to explore the connection between orbital behaviors of hyperspace dynamical systems and Choquet capacities of random closed sets. Moreover, relations between the hit-or-miss topology and other hyperspace topologies or metrics such as the Vietoris topology, Hausdorff metric and Hausdorff–Buseman metric are also given