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In this paper, a concept of nearness convergence is introduced which contains the proximal convergence of Leader as a special case. It is proved that uniform convergence and this nearness convergence are equivalent on totally bounded uniform nearness spaces. One of the features of this convergence is that it lies between uniform convergence and pointwise convergence, and implies uniform convergence on compacta. Some other weaker notions of nearness convergence which are sufficient to preserve nearness maps are also discussed.
The purpose of this paper is to develop a general technique for attacking problems involving extensions of continuous functions from dense subspaces and to use it to obtain new results as well as to improve some of the known ones. The theory of structures developed by Harris is used to get some general results relating filters and covers. A necessary condition is derived for a continuous function f: X → Y to have a continuous extension : λx → λy where λZ denotes a given extension of the space Z. In the case of simple extensions, is continuous and in the case of strict extensions is θ-continuous. In the case of strict extensions, sufficient conditions for uniqueness of are derived. These results are then applied to several extensions considered by Banaschewski, Fomin, Kattov, Liu-Strecker, Blaszczyk-Mioduszewski, Rudölf, etc.
We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.
In 1964 Frink  generalized Wallman's method  of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro , , Banasĉhewski , Njåstad  and Steiner ). Recently Alò and Shapiro ,  have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner .
This tract aims at providing a compact introduction to the theory of proximity spaces and their generalizations. It is hoped that a study of the tract will better enable the reader to understand the current literature. In view of the fact that research material on proximity spaces is scattered and growing rapidly, the need for such a survey is apparent. The material herein is self-contained except for a basic knowledge of topological and uniform spaces, as can be found in standard texts such as the one by John L. Kelley; in fact, for the most part, we use Kelley's notation and terminology.
The tract begins with a brief history of the subject. The first two chapters give the fundamentals and the pace of development is rather slow. We have tried to motivate definitions and theorems with the help of metric and uniform spaces; a knowledge of the latter is, however, not necessary in understanding the proofs. The main result in these two chapters is the existence of the Smirnov compactification, which is proved using clusters. Taking advantage of hindsight, several proofs have been considerably simplified.
A reader not acquainted with uniform spaces will find it necessary to become familiar with such spaces before reading the third chapter. In this chapter, the interrelationships between proximity structures and uniform structures are considered and, since proximity spaces are intermediate between topological and uniform spaces, some of the most exciting results are to be found in this part of the tract.
The germ of the theory of proximity spaces showed itself as early as 1908 at the mathematical congress in Bologna, when Riesz discussed various ideas in his ‘theory of enchainment’ which have today become the basic concepts of the theory. The subject was essentially rediscovered in the early 1950's by Efremovič when he axiomatically characterized the proximity relation ‘A is near B’ for subsets A and B of any set X. The set X together with this relation was called an infinitesimal (proximity) space, and is a natural generalization of a metric space and of a topological group. A decade earlier a study was made by Krishna Murti, Wallace and Szymanski concerning the use of ‘separation of sets’ as the primitive concept. In each case similar, but weaker, axioms than those of Efremovič were used. Efremovič later used proximity neighbourhoods to obtain an equivalent set of axioms for a proximity space and thereby an alternative approach to the theory.
Defining the closure of a subset A of X to be the collection of all points of X ‘near’ A, Efremovič showed that a topology can be introduced in a proximity space and that one thereby obtains, in fact, a completely regular (and hence uniformizable) space. He further showed that every completely regular space X can be turned into a proximity space with the help of Urysohn's function: namely, A § B iff there exists a continuous function f mapping X into such that f(A) = 0 and f(B) = 1.
This tract provides a compact introduction to the theory of proximity spaces and their generalisations, making the subject accessible to readers having a basic knowledge of topological and uniform spaces, such as can be found in standard textbooks. Two chapters are devoted to fundamentals, the main result being the proof of the existence of the Smirnov compactification using clusters. Chapter 3 discusses the interrelationships between proximity spaces and uniform spaces and contains some of the most interesting results in the theory of proximity spaces. The final chapter introduces the reader to several generalised forms of proximity structures and studies one of them in detail. The bibliography contains over 130 references to the scattered research literature on proximity spaces, in addition to general references.