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Abstract: We study covering properties of hyperspaces with the Vietoris topology, as well as with its lower and upper parts. Considered covering properties are defined in terms of selection principles: the Menger, Hurewicz and Rothberger properties, and γ-sets and their relatives. [Copyright &y& Elsevier]

Abstract: In this paper we investigate relationships between closure-type properties of hyperspaces over a space X and covering properties of X. [Copyright &y& Elsevier]

Abstract: The aim of this paper is twofold: first we investigate the role of proximity notions in the framework of approach theory [R. Lowen, Approach Spaces. The Missing Link in the Topology-Uniformity-Metric Triad, Clarendon Press, Oxford, 1997] and show how quantified proximity structures can be used to obtain a canonical intrinsic version of the Smirnov compactification in this setting. Second we introduce a hyperspace structure which commutes with this type of Smirnov compactification, solving a ‘dual’ problem to the one treated concerning approach completion in [R. Lowen, M. Sioen, Proximal hypertopologies revisited, Set-Valued Anal. 6 (1998) 1–19]. [Copyright &y& Elsevier]

We study properties related to first countability and countable compactness of uniformizable Δ-hit-and-miss hyperspace topologies. We show that they are proximal hyperspace topologies. We use the Smirnov compactification as a tool in our investigation. [Copyright &y& Elsevier]

Beer and Tamaki investigated necessary and sufficient conditions for the uniformizability of (proximal) Δ-topologies.Their proofs involved construction of special Urysohn functions. In this paper we attack the same problem using as a useful tool a uniform topology with reference to a Hausdorff uniformity patterned after the one related to the Attouch–Wets topology. We also study ΔU-topologies, proximal ΔU-topologies which are natural generalizations of the U-topology discovered by Costantini and Vitolo. [Copyright &y& Elsevier]