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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 412796, 10 pages
http://dx.doi.org/10.1155/2013/412796
Research Article

Strong Proximal Continuity and Convergence

1Department of Mathematics, Seconda Università degli Studi di Napoli, 81100 Caserta, Italy
2Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy
396 Dewson Street, Toronto, ON, Canada M3J 1P3

Received 16 October 2012; Accepted 9 January 2013

Academic Editor: Yuriy Rogovchenko

Copyright © 2013 Agata Caserta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.