First-countability, sequentiality and tightness of the upper Kuratowski convergence
type: Item
instance of: scholarly article
title: First-countability, sequentiality and tightness of the upper Kuratowski convergence
zbMATH Open document ID: 1064.54017
author: Frédéric Mynard
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-05-27T00:00:00Z
review text: Generalizing previous results by \textit{B. Alleche} and \textit{J. Calbrix} [Topology Appl. 93, 207--218 (1999; Zbl 0943.54013)] and \textit{C. Costantini, L. Holá} and \textit{P. Vitolo} [Topology Appl. 142, 245--292 (2004; Zbl 1065.54003)], the author characterizes the first countability of the upper Kuratowski convergence in the general context of convergences. As a corollary, he obtains the following result: Theorem 0.3. The following are equivalent: 1. X is hereditarily Lindelöf; 2. \([X,K]\) is first-countable; 3. \([X,K]\) is sequential; 4. \([X,K]\) is sequentially tight; 5. \(T[X,K]\) is sequential; 6. \(T[X,K]\) is countably tight. Here, \([X,K]\) denotes the associated upper Kuratowski convergence and \(T[X,K]\) denotes the associated upper Kuratowski topology. He also applies his results to consonance and generalized (real-valued) consonance results.
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 2067062
zbMATH Keywords: first countability
zbMATH Keywords: hyperspaces
zbMATH Keywords: upper Kuratowski convergence
zbMATH Keywords: sequentiality
zbMATH Keywords: tightness
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069940
OpenAlex ID: W4243572111
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DOI: 10.1216/RMJM/1181069940
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description: scientific article; zbMATH DE number 2067062