Properties related to first countability and countable compactness in hyperspaces: A new approach.
type: Item
instance of: scholarly article
title: Properties related to first countability and countable compactness in hyperspaces: A new approach.
zbMATH Open document ID: 1052.54013
DOI: 10.1016/S0166-8641(03)00201-3
author: Ľubica Holá
author: E. Meccariello
author: Giuseppe Di Maio
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: 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.
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 2056845
zbMATH Keywords: Hyperspace topology
zbMATH Keywords: (Proximal) \(\varDelta\)-hit-and-miss hyperspace topology
zbMATH Keywords: First countability
zbMATH Keywords: Tightness
zbMATH Keywords: Character
zbMATH Keywords: Strong countable compactness
zbMATH Keywords: \(\omega\)-boundedness
zbMATH Keywords: Uniformity
zbMATH Keywords: Proximity
zbMATH Keywords: Smirnov compactification
zbMATH Keywords: Embedding
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2056845