What distinguishes a subway map from other types of maps? Why does a knot become knotted? The Möbius strip is lopsided for what reason? These are topological inquiries, the mathematical study of the characteristics kept when things are bent or stretched. Topology expanded and became as basic in the 20th century as algebra and geometry, with significant ramifications for science, particularly physics.

In this Very Short Introduction, Richard Earl discusses the formal notion of continuity as well as some of the more visually appealing aspects of topology (looking at surfaces). He pays homage to the historical figures, issues, and surprises that have fueled the development of this area by taking into account some of the eye-opening cases that made mathematicians realize a necessity for studying topology.