Properties related to first countability and countable compactness in hyperspaces: A new approach.
Let \(X\) be a Hausdorff space and \(CL(X)\) be the hyperspace of all closed and nonempty subsets of \(X\). Given \(\Delta\subset CL(X)\), a hit-and-miss hypertopology \(\tau(\Delta)\) on \(CL(X)\) is the supremum of two topologies: the lower part, i.e., the lower Vietoris \(\tau^-_V\) (a subbasic open set of \(\tau^-_V\) consists of all closed sets that ``hit'' a given open subset of \(X\)) and the upper part \(\tau(\Delta)^+\) (a subbasic open set of \(\tau(\Delta)^+\) consists of all closed subsets of \(X\) that ``miss'' a given element of \(\Delta\)). When \(X\) is a metric space with metric \(d\) another upper part can be obtained, namely the upper proximal \(\Delta\)-hit-and-miss topology \(\sigma(\Delta,d)^+\) which is generated by all collections of subsets of \(X\) which ``really miss'' elements of \(\Delta\), i.e., have a positive distance apart from elements of \(\Delta\). In this case, the supremum of the topologies \(\tau^-_V\) and \(\sigma(\Delta,d)^+\) is the proximal \(\Delta\)-miss-and-hit topology. In this paper the authors study properties related to first countability as well as to compactness (like strong countable compactness, countable compactness) of uniformizable \(\Delta\)-hit-and-miss hyperspace topologies. They show that they are proximal hyperspace topologies. They use the Smirnov compactification as a tool in their investigation.
(Proximal) \(\varDelta\)-hit-and-miss hyperspace topology
scientific article; zbMATH DE number 2056845
scientific article