You are here

Essentials of Topology with Applications

Steven C. Krantz
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics
[Reviewed by
P. N. Ruane
, on

I’ve still got the copy of the book by Wofgang Thron that introduced me to general topology in 1966. It’s called Topological Structures and its 200 pages contain not one diagram. Remarkably, the whole theoretical edifice is illuminated by just one example, and the text mainly consists of definitions and theorems. Of course, the titles of such books often used to include the words ‘treatise’ or ‘foundations’; but Thron’s book is packed with historical commentary — in fact, it constitutes a fascinating evolutionary account of the development of point-set topology.

At the other extreme, much work used to be done (and could still be done) on visual topology in UK primary and secondary school levels. The approach was always investigational, and the topics included colouring problems, properties of Moebius strips (varying the number of twists, and slicing in various ways). There were exercises on traversiblity and Euler’s theorem for polyhedra, together with their associated networks. In fact, the relationship of the Moebius strip to the Klein bottle can be made visually plausible and the joke about the coffee cup and the donut was appreciated by eleven year old topologists. Unfortunately, all such work was abandoned long ago, because forces of educational conservatism demanded that more attention be paid to archaic topics such as long division.

So there we have the two seemingly irreconcilable aspects of topology. The first (which can mystify the math major) deals mainly with local results, and uses set-theoretic tools. The second aspect (which can be introduced at any educational level) deals mainly with geometrically motivated global results, and its eventual theoretical basis is largely algebraic.

Nowadays, however, many topology textbooks narrow the gap between general and algebraic topology, and this book by Stephen Krantz is successful in this respect. But initial perusal of its pages may lead one to think that it constitutes a ‘topology-made-easy’ approach. This illusion arises because the book is abundantly illustrated, the text is spaciously accommodated within its pages, and the chapter headings convey an exciting range of sub-themes and applications that were never included in topology books of yore.

The truth is, however, that this introduction to topology is as challenging as any other — and yet more readable than most. The first two chapters cover more than is needed for a first course in point set topology, and a wide range of examples are invoked. Apart from familiar examples, such as the discrete topology, the topology of finite complements and the usual topology, there is reference to the topology of the Moore plane, the long line, the Sorgenfrey topology and the box topology. The contexts in which ideas are discussed are both analytic and geometric. For example, there is a delightful introduction to Morse theory based upon the examination of smooth height functions on the torus, and there is an application of those pesky separation axioms to the topology of digital imaging.

Having said this, there are features of point-set topology that are perennial sources of frustration. For example, many intuitively obvious results require proofs that involve tricky logical arguments (e.g. the fact that the closure of S is the union of S and ∂S). And things like the Tychonoff theorems remind me of Cantor’s comment to Dedekind, paraphrased as: “I follow it, but I don’t see it.” Well, even the most expert of expositors can’t resolve matters of inherent mathematical inscrutability — nor can they compensate for the fact that opacity is often in the eye of the beholder.

Chapter 3, in less than 60 pages, illuminates the two basic aspects of algebraic topology: homotopy, singular homology and the relation between them. In fact, the introduction to homotopy is lucidly concise, and I’ve never seen a more accessible proof that π1(S1) = Z. On the other hand, I found the section on singular homology a greater challenge, possibly because I can’t easily think in terms of N-chains. Yet there are some familiar (but still wonderful) results here, such as the fact that the first homology group is the abelianization of the first homotopy group. The treatment of algebraic topology concludes with an account of the powerful methods of covering spaces and the concept of index.

Although the book’s title refers to the “essentials” of topology, the author suggests that, after the very short 4th chapter (Manifold Theory), the remaining five chapters can be investigated as interest dictates. There is a very short chapter called Moore-Smith Convergence, followed by longer chapters on Function Spaces, Dynamical Systems and highly readable introductions to Knot Theory and Graph Theory.

Two other aspects of this book that deserve mention are the exercises and the seventy pages of appendices. There are about 250 exercises whose dual purpose is to consolidate and extend previously introduced ideas. Most of these are challenging and none are simplistic. There are solutions to about two-thirds of the exercises.

The appendices consist of background material on elementary logic, set theory, the real numbers, the axiom of choice and very brief notes on groups, vector spaces etc. But I feel that, if readers need to be refreshed in such very basic ideas, they may not be able to cope with the demands of the early chapters in this book. The minimum pre-requisite knowledge should be a good grounding in real analysis. In fact, an ideal preparation would consist of familiarity with the contents of the author’s excellent Guide to Real Variables.

Reading this book, I discovered only one (very obvious) misprint (p. 10), and two places where the notation could have been more clearly defined (Defn. 3.2.1 and the symbol “t” on p. 201). Other than that, the book has been written with a high degree of accuracy. Its unusual structure, together with its attractive appearance and reader-friendly narrative, means that Steven Krantz’s latest work is highly recommended as the basis for a first course in topology.

Peter Ruane has taught topology to students in the age-range 5 to 50.


What Is Topology?

First Definitions


The Separation Axioms






Totally Disconnected Spaces

The Cantor Set

Metric Spaces


Baire’s Theorem

Lebesgue’s Lemma and Lebesgue Numbers

Advanced Properties of Topological Spaces

Basis and Sub-Basis

Product Spaces

Relative Topology

First Countable, Second Countable, and So Forth


Quotient Topologies


Morse Theory

Proper Mappings


An Application to Digital Imaging

Basic Algebraic Topology

Homotopy Theory

Homology Theory

Covering Spaces

The Concept of Index

Mathematical Economics

Manifold Theory

Basic Concepts

The Definition

Moore–Smith Convergence and Nets

Introductory Remarks


Function Spaces

Preliminary Ideas

The Topology of Pointwise Convergence

The Compact-Open Topology

Uniform Convergence

Equicontinuity and the Ascoli–Arzela Theorem

The Weierstrass Approximation Theorem

Knot Theory

What Is a Knot?

The Alexander Polynomial

The Jones Polynomial

Graph Theory


Fundamental Ideas of Graph Theory

Application to the Königsberg Bridge Problem

Coloring Problems

The Traveling Salesman Problem

Dynamical Systems


Planar Autonomous Systems

Lagrange’s Equations

Appendix 1: Principles of Logic


"And" and "Or"


"If - Then"

Contrapositive, Converse, and "Iff"


Truth and Provability

Appendix 2: Principles of Set Theory

Undefinable Terms

Elements of Set Theory

Venn Diagrams

Further Ideas in Elementary Set Theory

Indexing and Extended Set Operations

Countable and Uncountable Sets

Appendix 3: The Real Numbers

The Real Number System

Construction of the Real Numbers

Appendix 4: The Axiom of Choice and Its Implications

Well Ordering

The Continuum Hypothesis

Zorn’s Lemma

The Hausdorff Maximality Principle

The Banach–Tarski Paradox

Appendix 5: Ideas from Algebra





Vector Spaces

Solutions of Selected Exercises



Exercises appear at the end of each chapter.