First-countability, sequentiality and tightness of the upper Kuratowski convergence
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.
scientific article; zbMATH DE number 2067062
scientific article