Strong Proximal Continuity and Convergence

Agata Caserta,Roberto Lucchetti,Som Naimpally

doi:10.1155/2013/412796

Agata Caserta, Roberto Lucchetti + Show 1 more

Open Access

https://doi.org/10.1155/2013/412796

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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.

Highlights

In topology and analysis the concepts of uniform continuity and uniform convergence on compacta play a central role
After revising the various concepts of continuity in uniform and proximal spaces, we introduce several forms of strong proximal convergences, and we investigate their connections in the setting of Tychonoff spaces with compatible uniformities and proximities
We compare them with uniform convergences, and we study them on bornologies

Summary

Introduction

In topology and analysis the concepts of uniform continuity and uniform convergence on compacta play a central role. In Beer and Levi [1] and Caserta et al [2] a definition of strong uniform convergence is considered, enjoying the property of preserving strong uniform continuity This concept is related to the sticking convergence defined by Bouleau [3] (see [4]). Beer and Levi [1] introduced some new concepts of local-type proximal continuity and local-type proximal convergence, in the setting of metric spaces, and provided some characterizations of strong uniform continuity and strong uniform convergence in terms of the analogous proximal properties. These characterizations rely on convergence of various types on bornologies. Our results generalize similar ones in Beer and Levi [1] and Di Maio et al [11,12,13]

Preliminaries

Convergences

Convergence on Hyperspaces