4. The plane and other spaces

Most functions have several numerical inputs and produce more than one numerical output. But even generally continuity requires that we can constrain the difference in outputs by suitably constraining the difference in inputs. ‘The plane and other spaces’ asks more general questions such as ‘is the distance a car has travelled a continuous function of its speed?’ This is a subtle question as neither the input nor output are numbers, but rather functions of time, with input the speed function s(t) and output the distance function d(t). In answering the question, it considers continuity between metric spaces, equivalent metrics, open sets, convergence, and compactness and connectedness, the last two being topological invariants that can be used to differentiate between spaces.

Epilogue

Topology remains a large, active research area in mathematics. Unsurprisingly its character has changed over the last century—there is considerably less current interest in general topology, but whole new areas have emerged, such as topological data analysis to help analyze big data sets. The Epilogue concludes that the interfaces of topology with other areas have remained rich and numerous, and it can be hard telling where topology stops and geometry or algebra or analysis or physics begin. Often that richness comes from studying structures that have interconnected flavours of algebra, geometry, and topology, but sometimes a result, seemingly of an entirely algebraic nature say, can be proved by purely topological means.

5. Flavours of topology

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.

3. Thinking continuously

Many topologists might choose to describe their subject as the study of continuity. There are continuous and discontinuous functions in our everyday routines. ‘Thinking continuously’ aims to provide a more rigorous sense of what continuity entails for real-valued functions of a real variable. It focuses on functions having a single numerical input and a single numerical output. The properties of continuous functions are considered and the boundedness theorem and intermediate value theorem are also explained.

2. Making surfaces

‘Making surfaces’ considers the shape of surfaces and discusses the work of some of the early topologists, Möbius, Klein, and Riemann. It introduces the torus shape and shows how its Euler number can be calculated along with that of a sphere. It discusses closed surfaces—ones without a boundary—and how they can be divided up into vertices, edges, and faces. It then introduces one-sided surfaces such as the Möbius strip and Klein bottle, which are examples of non-orientable surfaces. The Euler number goes a long way to separating out different surfaces, with the only missing ingredient in the classification the notion of orientability.

6. Unknot or knot to be?

‘Unknot or knot to be?’ explains that a knot is a smooth, simple, closed curve in 3D space. Being simple and closed means the curve does not cross itself except that its end returns to its start. All knots are topologically the same as a circle; what makes a circle knotted—or not—is how that circle has been placed into 3D space. The central problem of knot theory is a classification theorem: when is there an ambient isotopy between two knots or how do we show that no such isotopy exists? Key elements of knot theory are discussed, including the three Reidemeister moves, prime knots, adding knots, and the Alexander and Jones polynomials.

1. What is topology?

Topology is now a major area of modern mathematics, but an appreciation of topology came late in the history of mathematics. The word topology—meaning ‘the study of place’—was not coined until 1836. ‘What is topology?’ aims to provide a sense of topology’s ideas and its technical vocabulary. It discusses the concepts of letters being topologically the same or homeomorphic and then moves on to Euler’s formula, which shows that there are only five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Early problems in topology included defining what dimension means and point-set topology, which sought to address what it means to be a set or to be a space.