Abstract and Applied Analysis

Strong Proximal Continuity and Convergence

Agata Caserta, Roberto Lucchetti, and Som Naimpally

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Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 412796, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511785

Digital Object Identifier
doi:10.1155/2013/412796

Mathematical Reviews number (MathSciNet)
MR3035366

Zentralblatt MATH identifier
1287.54022

Citation

Caserta, Agata; Lucchetti, Roberto; Naimpally, Som. Strong Proximal Continuity and Convergence. Abstr. Appl. Anal. 2013 (2013), Article ID 412796, 10 pages. doi:10.1155/2013/412796. https://projecteuclid.org/euclid.aaa/1393511785


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