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2013 Strong Proximal Continuity and Convergence
Agata Caserta, Roberto Lucchetti, Som Naimpally
Abstr. Appl. Anal. 2013: 1-10 (2013). DOI: 10.1155/2013/412796


In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself.


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Agata Caserta. Roberto Lucchetti. Som Naimpally. "Strong Proximal Continuity and Convergence." Abstr. Appl. Anal. 2013 1 - 10, 2013.


Published: 2013
First available in Project Euclid: 27 February 2014

zbMATH: 1287.54022
MathSciNet: MR3035366
Digital Object Identifier: 10.1155/2013/412796

Rights: Copyright © 2013 Hindawi

Vol.2013 • 2013
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