This book is based on a course taught to an audience of undergraduate and graduate students at Oxford, and can be viewed as a bridge between the study of metric spaces and general topological spaces. About half the book is devoted to relatively little-known results, much of which is published here for the first time. The author sketches a theory of uniform transformation groups, leading to the theory of uniform spaces over a base and hence to the theory of uniform covering spaces. Readers interested in general topology will find much to interest them here.

Exceptionally smooth, clear, detailed examination of uniform spaces, topological groups, topological vector spaces, topological algebras and abstract harmonic analysis. Also, topological vector-valued measure spaces as well as numerous problems and examples. For advanced undergraduates and beginning graduate students. Bibliography. Index.

The principal aim of this book is to introduce topology and its many applications viewed within a framework that includes a consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising.It is the first complete presentation on topology with applications considered in the context of proximity spaces, and the nearness and remoteness of sets of objects. A novel feature throughout this book is the use of near and far, discovered by F Riesz over 100 years ago. In addition, it is the first time that this form of topology is presented in the context of a number of new applications.

Uniform spaces play the same role for uniform continuity as topological spaces for continuity. The theory was created in 1936 by A. Weil, whose original axiomatization was soon followed by those of Bourbaki and Tukey; in this book use is made chiefly of Tukey's system, based on uniform coverings. The organization of the book as a whole depends on the Eilenberg-MacLane notions of category, functor and naturality, in the spirit of Klein's Erlanger Program but with greater reach. The preface gives a concise history of the subject since 1936 and a foreword outlines the category theory of Eilenberg and MacLane. The chapters cover fundamental concepts and constructions; function spaces; mappings into polyhedra; dimension (1) and (2); compactifications and locally fine spaces. Most of the chapters are followed by exercises, occasional unsolved problems, and a major unsolved problem; the famous outstanding problem of characterizing the Euclidean plane is discussed in an appendix. There is a good index and a copious bibliography intended not to itemize sources but to guide further reading.

Topological Spaces focuses on the applications of the theory of topological spaces to the different branches of mathematics. The book first offers information on elementary principles, topological spaces, and compactness and connectedness. Discussions focus on locally compact spaces, local connectedness, fundamental concepts and their reformulations, lattice of topologies, axioms of separation, fundamental concepts of set theory, and ordered sets and lattices. The manuscript then ponders on mappings and extensions and characterization of topological spaces, including completely regular spaces, transference of topologies, Wallman compactification, and embeddings. The publication takes a look at metric and uniform spaces and applications of topological groups. Topics include the Stone-Weierstrass Approximation Theorem, extensions and completions of topological groups, topological rings and fields, extension and completion of uniform spaces, uniform continuity and uniform convergence, metric spaces, and metritization. The text is a valuable reference for mathematicians and researchers interested in the study of topological spaces.

This book is based on lectures I have given to undergraduate and graduate audiences at Oxford and elsewhere over the years. My aim has been to provide an outline of both the topological theory and the uniform theory, with an emphasis on the relation between the two. Although I hope that the prospec tive specialist may find it useful as an introduction it is the non-specialist I have had more in mind in selecting the contents. Thus I have tended to avoid the ingenious examples and counterexamples which often occupy much ofthe space in books on general topology, and I have tried to keep the number of definitions down to the essential minimum. There are no particular pre requisites but I have worked on the assumption that a potential reader will already have had some experience of working with sets and functions and will also be familiar with the basic concepts of algebra and analysis. There are a number of fine books on general topology, some of which I have listed in the Select Bibliography at the end of this volume. Of course I have benefited greatly from this previous work in writing my own account. Undoubtedly the strongest influence is that of Bourbaki's Topologie Generale [2], the definitive treatment of the subject which first appeared over a genera tion ago.

gentle introduction to the subject, leading the reader to understand the notion of what is important in topology with regard to geometry. Divided into three sections - The line and the plane, Metric spaces and Topological spaces -, the book eases the move into higher levels of abstraction. Students are thereby informally assisted in learning new ideas while remaining on familiar territory. The authors do not assume previous knowledge of axiomatic approach or set theory. Similarly, they have restricted the mathematical vocabulary in the book so as to avoid overwhelming the reader, and the concept of convergence is employed to allow students to focus on a central theme while moving to a natural understanding of the notion of topology. The pace of the book is relaxed with gradual acceleration: the first nine sections form a balanced course in metric spaces for undergraduates while also containing ample material for a two-semester graduate course. Finally, the book illustrates the many connections between topology and other subjects, such as analysis and set theory, via the inclusion of "Extras" at the end of each chapter presenting a brief foray outside topology.