Somashekhar Naimpally’s research while affiliated with Universidad de la Cañada and other places

This article gives a brief overview of Somashekhar(Som) Naimpally’s life and work. Som received his Ph.D. in Mathematics from Michigan State University in 1964 (a fairly recent picture of Som is shown on this page). The focus of his doctoral research was on essential fixed points and almost continuous functions, leading to his work on graph topology for function spaces, eventually leading to his work on the extension of continuous functions and various forms of proximity spaces. His supervisor was John G. Hocking, whose life and work were commemorated by Som.

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.

The focus of this special issue of Mathematics in Computer Science is near set theory and applications. The study of various forms of nearness relations in proximity space theory and the penultimate notions of near and far in topology span over 100 years, starting with an address by F. Riesz at the 1908 ICM congress in Rome.

The main purpose in writing this book is to demonstrate the beneficial use of near and far, discovered by F. Riesz over a century ago, from the undergraduate to the research level in general topology and its applications. Use of near and far is intuitive yet rigourous at the same time, which is rare in mathematics. The near and far paradigm is based on many years of teaching and research by the authors. It includes a presentation of spatially near sets as well as descriptively near sets. The book introduces topology and its many applications viewed within a framework that begins with metric spaces and deals with the usual topics such as continuity, open and closed sets, metric nearness, compactness and completeness and glides into topology, proximity and uniformity. Most topics are first studied in metric spaces and later in a topological space. The motivation for this approach is a straightforward, intuitive and yet rigourous rendition of topological concepts. The approach also unifies several scattered results in many areas. Many exercises come from the current literature and some occur in simplified form in metric spaces. The end result is a solid, workable framework for the study of topology with a variety of applications in science and engineering that include camouflage, cell biology, digital image analysis, forgery detection, general relativity, microscopy, micropalaeontology, pattern recognition, population dynamics, psychology and visual merchandising. We gratefully acknowledge the insightful Foreword by I.A. Taimanov. This Foreword is especially important, since it lucidly brings together the principal highlights of this book and it serves as a commemoration of the seminal work on topology by A.D. Taimanov, who proved one of the most fundamental results in topology concerning extensions of continuous functions from dense subspaces.

This survey paper contains a brief account of the literature on preservation of continuity under various convergences from the beginning to current research. We include stronger forms of continuity such as uniform continuity, proximal continuity, strong uniform continuity and weaker forms of continuity such as quasi-continuity, near continuity and almost continuity. A few proofs of the results are given and references are given for others. A standard reference for this research area is R.A. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lecture Notesin Mathematics 1315, Springer, 1988.

We produce necessary and sufficient conditions for the pointwise product of two uniformly continuous real-valued functions defined on a metric space to be uniformly continuous.

This article considers the problem of how to formulate a framework for the study of the nearness of collections of subsets of a set (also more tersely termed families of a set). The solution to the problem stems from recent work on approach spaces, near sets, and a specialised form of gap functional. The collection of all subsets of a set equipped with a distance function is an approach space.

The subject of hyperspace topologies on closed or closed and compact subsets of a topological space X began in the early part of the last century with the discoveries of Hausdorff metric and Vietoris hit-and-miss topology. In course of time, several hyperspace topologies were discovered either for solving some problems in Applied or Pure Mathematics or as natural generalizations of the existing ones. Each hyperspace topology can be split into a lower and an upper part. In the upper part the original set inclusion of Vietoris was generalized to proximal set inclusion. Then the topologization of the Wijsman topology led to the upper Bombay topology which involves two proximities. In all these developments the lower topology, involving intersection of finitely many open sets, was generalized to locally finite families but intersection was left unchanged. Recently the authors studied symmetric proximal topology in which proximity was used for the first time in the lower part replacing intersection with its generalization: nearness. In this paper we use two proximities also in the lower part and we obtain the lower Bombay hypertopology. Consequently, a new hypertopology arises in a natural way: the symmetric Bombay topology which is the join of a lower and an upper Bombay topology.

We offer a unified approach to investigate hypertopologies. In this setting the proofs are simple and transparent. New problems are raised.