Topology: A Very Short Introduction

Citation

Earl, R. (2019). Topology: A Very Short Introduction. Oxford University Press. Very Short Introductions.

Chapter Summary

The book belongs to Oxford's pocket-sized Very Short Introductions series and has seven chapters, followed by a historical timeline, suggestions for further reading and an index. The publisher frames the subject through three questions (how a subway map differs from other maps, what makes a knot knotted, and what makes the Möbius strip one-sided) and describes topology as the study of properties preserved when objects are twisted or stretched, a field that in the twentieth century became as broad and fundamental as algebra and geometry. The author is Director of Undergraduate Studies at Oxford's Mathematical Institute and has taught topology at undergraduate and graduate level as well as to secondary-school students. According to the publisher, the book combines the visual side of topology, especially surfaces, with the formal definition of continuity, and pays attention to the people, problems and surprising examples that made the subject necessary.

Chapter 1: What is Topology?

The opening chapter sets out what topology studies and why: the properties of shapes that survive stretching and twisting, the questions about maps, knots and one-sided surfaces with which the publisher introduces the book.

Chapter 2: Making Surfaces

The first of two chapters on surfaces, the visual heart of the subject that the publisher emphasises: how surfaces can be made and recognised.

Chapter 3: Thinking Continuously

Continuity, the central idea of topology. The publisher notes that the book covers the formal definition of continuity, the step that turns intuitive deformation into mathematics, and this is the chapter devoted to the idea.

Chapter 4: The Plane and Other Spaces

The chapter moves from the familiar plane to other spaces in which continuity makes sense, widening the setting of Chapter 3.

Chapter 5: Flavours of Topology

A survey of the different branches into which topology has grown, in keeping with the publisher's point that it became as broad a field as algebra and geometry.

Chapter 6: More on Surfaces

A return to surfaces, now with the tools of the intervening chapters available.

Chapter 7: Knot to Be

Knots, the subject of the last of the publisher's opening questions: what makes a knot knotted.

Key Concepts

Chapters 1–2: Topological thinking and surfaces

• Topological property: a feature of a shape that is unchanged by stretching or twisting without tearing, the kind of information a subway map keeps while discarding distances.
• Surface: a two-dimensional shape such as the Möbius strip; surfaces are the main visual examples of the subject.
• One-sidedness: the property of the Möbius strip that a single side runs all the way around it.

Chapters 3–4: Continuity and spaces

• Continuity: the formal expression of the idea that nearby points stay nearby, the basic notion on which topology rests.
• Spaces beyond the plane: settings more general than the plane in which continuity can still be defined.

Chapters 5–7: Branches, surfaces and knots

• Branches of topology: the different directions, visual and abstract, in which the subject has developed.
• Knots: closed loops in space, and the question of when one can be deformed into another.

Critical Analysis

Strengths

• Visual and formal together. According to the publisher, the book gives a sense of the visual side of topology and also covers the formal definition of continuity, so readers see both the intuition and how mathematicians make it precise. The contents reflect that balance, with chapters on surfaces and knots on either side of the chapters on continuity and spaces.
• Written by an experienced teacher. The author has taught topology at university level and to school students, and is also the author of Towards Higher Mathematics: A Companion.
• History as part of the story. The publisher stresses the historical people, problems and surprises behind the subject, and the book includes a historical timeline as well as further reading.
• Covers both surfaces and knots. Surfaces (two chapters) and knots (the final chapter) give a reader with no background concrete examples to hold on to.

Limitations

• Very short by design. Seven chapters in a pocket book can only introduce each topic; the contents list no exercises, and it is not a course text.
• Some abstraction. Because the book includes the formal definition of continuity, readers expecting a purely pictorial account may find Chapters 3 and 4 more demanding than the chapters on surfaces and knots.

Where It Fits in the Course

This is the course's shortest way into topology, to read first. Jeffrey Weeks's The Shape of Space is the natural next book: it is still informal and visual, but it is a textbook with exercises and answers that goes much further into surfaces, curvature and three-dimensional spaces, ending with the shape of the universe. Read Earl for a quick map of the whole subject and Weeks for a deeper understanding of its geometric examples.

The formal books come afterwards. Wilson Sutherland's Introduction to Metric and Topological Spaces develops continuity through metric spaces before generalising to topological spaces, the formal version of the ideas of Earl's Chapters 3 and 4; James Munkres's Topology is a full course in point-set topology followed by the fundamental group; Allen Hatcher's Algebraic Topology develops one major branch of the subject, the fundamental group, homology and cohomology, in full. Earl's book makes a good overview to reread before starting any of them.

How to Use This Book

The book is written for general readers and is a good fit for school students thinking about a mathematics degree, first-year undergraduates who have heard of topology but not studied it, and students in other subjects who want to know what it is about. Secondary-school mathematics is enough; the formal material in the middle chapters needs patience more than background.

Read it straight through in a few sittings. Chapters 1 and 2 give the intuitive picture; take Chapters 3 and 4 slowly, since they hold the formal idea of continuity on which the rest of the subject is built; Chapters 5–7 can then be read for enjoyment. Use the historical timeline to place the people and results the book mentions.

As you read, make the shapes: cut and tape a Möbius strip, draw the surfaces of Chapters 2 and 6, and tie a few knots to try to tell them apart. Then use the further reading list, or move on to Weeks's The Shape of Space, to work through the ideas with exercises.