If you were to write a topology book, what would be the chapter index? Which theorems would you absolutely include?

This isn’t the first time I’m citing Stillwell’s opening paragraph of Classical Topology and Combinatorial Group Theory:

In recent years, many students have been introduced to topology in high school mathematics. Having met the Möbius band, the seven bridges of Königsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still doesn’t understand the simplest topological facts, such as the reason why knots exist.

I’m bringing this up to make the same point made by Andrew Winkler in his answer: “Topology” means different things to different people, even if we’re interpreting “topology book” as “an undergrad textbook suitable for a first course on the subject”.

So: if I were to write a topology book, the first question I would ask myself is – who am I writing this for? And why? And what do I mean by “topology”?

Here’s the syllabus of Math 432 (Introduction to Topology) at the University of Maryland:

Metric Spaces, Topological Spaces

Continuous maps and homeomorphisms

Connectedness, compactness (including Heine-Borel, Bolzano-Weierstrass, Ascoli-Arzela theorems)

Cantor sets

Fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem)

Surfaces (Euler characteristic, the index of a vector field, hairy sphere theorem)

Elements of combinatorial topology (graphs and trees, planarity, coloring problems)

For my taste this is actually an excellent introductory course. There’s some “abstract” point-set topology but also a whole lot of what Stillwell is advocating for, and a nice selection of very cool applications. I’m a little surprised by the inclusion of Arzelà-Ascoli; it’s an important theorem but hard to motivate without some applications that would be hard to navigate to (in complex analysis, or ODEs). Cantor sets are a little bit of an outlier here but I would submit that everyone needs to know about them, so why not.

In short, this syllabus would make a nice table of contents for a topology textbook. Since a book can do more than a one-semester course I would probably add an introduction to homology and maybe cohomology, I would strengthen the combinatorial topology part with some knot theory, maybe(?) add a bit of Bass-Serre theory, and maybe also the Hopf fibration as a non-trivial example in the homotopy groups of spheres. This isn’t very coherent but it’s a lot of cool stuff.

As another example, here’s the (current) syllabus of Math 144 (Introduction to Topology and Geometry) at Stanford:

Point set topology, including connectedness, compactness, countability and separation axioms

The inverse and implicit function theorems

Smooth manifolds, immersions and submersions, embedding theorems

This is also a very nice syllabus but it has very different goals. The topology part is purely point-set, and then we move to the differentiable and smooth categories, probably to satisfy the promise of “Geometry” in the title. You can make an excellent textbook out of these headlines, too.

So, short answer: I don’t know. Why am I writing a topology book?

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