Topology

A book in 20 chapters about shape when stretching is allowed: from the bridges of Königsberg, Euler's formula and the Möbius strip to open sets, compactness and connectedness, the classification of surfaces, the fundamental group and homology, knots, and the Poincaré conjecture. Each chapter starts from a question, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.

How to read it

• In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
• Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
• Pictures first. Part I needs no algebra, only drawings and counting. Part II uses the language of sets (Logic and foundations, chapters 5 and 6) and builds on the metric spaces of Calculus and analysis, chapter 20.
• Groups from Part IV. Algebraic topology turns shapes into groups; the Algebra book's group chapters are the background.
• Rubber, not rigid. In Geometry, lengths and angles matter. Here only what survives bending and stretching counts, which is why a coffee cup and a doughnut are the same.

Contents

Part	Chapter	Level
I. Shapes up to stretching	1. What topology is	beginner
	2. Surfaces and Euler characteristic	beginner
	3. Graphs and planarity	beginner to intermediate
II. Point-set topology	4. Topological spaces	intermediate
	5. Continuity and homeomorphism	intermediate
	6. Building new spaces	intermediate
	7. Connectedness	intermediate
	8. Compactness	intermediate to advanced
	9. Separation and metrisation	advanced
III. Surfaces and manifolds	10. Classification of surfaces	intermediate to advanced
	11. Manifolds	advanced
IV. Algebraic topology	12. Homotopy	advanced
	13. The fundamental group	advanced
	14. The circle and covering spaces	advanced
	15. Van Kampen and surfaces	advanced to expert
	16. Homology	advanced to expert
	17. Applications of algebraic topology	advanced to expert
V. Frontier	18. Knot theory	advanced
	19. The Poincaré conjecture	expert
	20. Where topology goes	expert

The glossary defines every term, with a link to the chapter that teaches it.

Three ways in

• For the fun of shapes: chapters 1 to 3, then 10 and 18. Euler, the Möbius strip, the four colour theorem, every surface there is, and knots.
• For a first university course: chapters 4 to 9 (point-set topology), then 12 to 14.
• For the big theorems: chapters 13 to 17, then 19 and 20.