Proximity Spaces (Cambridge Tracts in Mathematics, 59) - Compact

In this review of Proximity Spaces (Cambridge Tracts in Mathematics, Series Number 59) the bottom line is straightforward: this short tract is best for readers who already know basic topology or uniform spaces and want a focused, rigorous introduction to proximity theory. The book's clearest achievement is presenting the fundamentals and the Smirnov compactification proof in a compact, readable form, making it a useful bridge between textbook background and research literature. For those seeking an accessible yet mathematically precise treatment of proximity structures, this tract delivers a concentrated course in undergraduates-to-graduate-ready language.

Key Features

• Compact introduction: Presents the core ideas of proximity spaces in a concise format that fits into a single tract while remaining mathematically complete.
• Foundational chapters: Two chapters focus on fundamentals, giving readers a clear development of definitions and basic properties useful for further study.
• Smirnov compactification proof: Contains a self-contained proof of the existence of the Smirnov compactification using clusters, valuable for students of topology.
• Connections to uniform spaces: Chapter 3 explores interrelationships with uniform spaces and highlights several important and interesting results in the theory.
• Generalised proximity structures: The final chapter introduces and examines several generalisations, offering pathways to current research topics.
• Bibliographic depth: Includes a bibliography with over 130 references, providing a clear map to the scattered research literature on proximity spaces.

Who It's For

The tract is aimed at advanced undergraduates, graduate students, and researchers who have a basic grounding in topological and uniform spaces and want a focused entry point into proximity theory without wading through a multi-volume treatise. It suits readers who appreciate concise, proof-oriented exposition and who plan to consult the cited research literature for deeper study.

It is less suitable for complete beginners in topology who lack familiarity with uniform spaces and standard textbook material, and it is not intended as a general-audience popular explanation; those readers should look for a broader introductory topology textbook before approaching this tract.

Pros & Cons

Pros

• Concise, rigorous presentation that makes the subject accessible to readers with prior topological background.
• Includes a clear proof of the Smirnov compactification using clusters, a helpful technical result for students and researchers.
• Shows useful links between proximity and uniform spaces, illuminating several interesting theorems.
• Extensive bibliography directs readers to over 130 research references for follow-up study.

Cons

• Not aimed at beginners lacking basic topological or uniform space knowledge, so newcomers may find it terse.

Specifications

Title	Proximity Spaces (Cambridge Tracts in Mathematics, 59)
Author	S. A. Naimpally
Series	Cambridge Tracts in Mathematics, Series Number 59
Focus	Theory of proximity spaces and generalisations
Notable content	Smirnov compactification proof using clusters
Bibliography	Over 130 references to research literature

Our Verdict

Proximity Spaces is a compact, well-focused tract that rewards readers who already know basic topology or uniform spaces; its clear proofs and the Smirnov compactification treatment make it a good value for students and researchers seeking a concise entry to proximity theory and pointers into the wider literature.

Frequently Asked Questions

Do I need prior topology to read this tract?
Yes. The tract assumes basic knowledge of topological and uniform spaces as typically found in standard textbooks.

Does the book include proofs of major results?
Yes. It contains detailed proofs, including a proof of the Smirnov compactification using clusters.

Is this suitable as a textbook for a course?
It can serve as a concise course supplement or seminar reading for advanced undergraduates or graduate students but is not a general introductory textbook.