Thoughts on topology

As well as Advanced Combinatorics, I have been teaching Topology this semester. This is something I had not taught for many years. I was only teaching half the module: the first half had been given, and the notes prepared, by James Mitchell. In this situation, and not expecting to be teaching the course again in the near future, I didn’t want to make big changes to the course. But it did provoke a few thoughts.

Products and Tychonoff’s Theorem

Products of topological spaces were discussed without any reference to the Axiom of Choice. I take the view that the Axiom of Choice is part of the general culture of mathematics: every student should be exposed to it, and should think about it.

As Bertrand Russell noted, the Axiom of Choice is precisely the statement that the Cartesian product of any family of non-empty sets is non-empty: this is why he called it the “multiplicative axiom”. (You no doubt recall his example: if a drawer contains infinitely many pairs of socks, how do you choose one sock from each pair?) So without it, the theory of products of topological spaces does rather fall apart. Of course it is true that most of the spaces which are important in topology can be seen to be non-empty without using AC. For example, finite products (including Euclidean spaces), or powers of a single set (the Cantor or Baire spaces).

Tychonoff’s Theorem states that a product of compact topological spaces is compact. This is not an easy theorem, and requires the Axiom of Choice in its proof (indeed, I believe it is equivalent to the Axiom of Choice). I saw a proof when I was a student, but I couldn’t find my old notes or textbook, and so I had to resort to Google. The final strategy I adopted in the lectures was as follows.

  • I began with two examples. The closed unit square is compact. This can be proved by observing that a space-filling curve is a continuous bijective function from the unit interval to the unit square. We had already proved that a continuous image of a compact space is compact. Also, the Cantor space, the product of countably many copies of {0,1} (with the discrete topology) is compact. This can be shown directly, by using the “middle third” construction of the Cantor space, which exhibits it as the complement of a collection of open subintervals of the closed unit interval, and hence closed; we had already proved that a closed subspace of a compact space is compact, and that the closed unit interval is compact (the Heine–Borel theorem, our motivating example for compactness).
  • By definition, a space is compact if and only if every open cover has a finite subcover. A straightforward argument shows that a space with a basis B for the topology is compact if and only if any cover by sets from B has a finite subcover. Now it is not too big a stretch to believe that a space with a sub-basis S for the topology is compact if and only if any cover by sets from S has a finite subcover. However, this is more difficult: it is Alexander’s sub-basis theorem, and uses the Axiom of Choice.
  • Now a sub-basis for the topology on a product space is given by the collection of inverse images of open sets in the factors under the projection maps onto those factors. Using this and Alexander’s theorem, it is not to hard to prove Tychonoff’s theorem.

In my time as a student, I was often asked to take the Jordan curve theorem on trust, and often promised that I would see a proof later; I never did. I think my little act of concealment was no worse than this.

The Baire Category Theorem

The course did not have much about metric spaces, except as examples of topological spaces (motivating the Hausdorff property, for example, or easing the transition from continuous functions on the real numbers to the general definition of continuous function).

As a result, the theorem of topology which I have used far more than any other, the Baire category theorem, was not in the course. (This theorem states that, in a complete metric space, the intersection of countably many open dense sets is non-empty.)

In the last lecture of the course, which I discuss below, I stated it, and gave a very simple application: a proof of the existence of transcendental numbers. Of course, most of my applications are in ultrametric spaces, especially spaces of paths in an infinite tree (where the proof is also much easier than in the general case). For example, Fraïssé limits exist because homogeneity is a countable sequence of requirements on a countable structure, each requirement being the restriction to an open dense subset. So for example, graphs isomorphic to the random graph are residual among all countable graphs.

Whither topology?

For the final lecture, one of the students had asked if I could say a bit about further directions in topology and in its applications to other parts of mathematics. This was, for me, not the usual undergraduate lecture: I talked about manifolds, algebraic topology (both homology and homotopy) and the Poincaré conjecture, differential topology, point-set topology, the Zariski topology in algebraic geometry, topological groups, and the Baire category theorem (as noted above). A packed 50 minutes, and I hope that the students took something away from the lecture!

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About Peter Cameron

I count all the things that need to be counted.
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4 Responses to Thoughts on topology

  1. Another thing I forgot to mention.
    The students know about graphs, so I told them about Henry Whitehead’s comment “Combinatorics is the slums of topology”, and invited them to consider the relationship between connectedness for graphs and connectedness for topological spaces. This was by way of introduction to path-connectedness, and the proof that it implies connectedness.

  2. jessemckeown says:

    I do not think the space-filling curves are bijective; happily, “surjective” suffices for compactness.

    • Of course you are right, sorry. Indeed, the students have an exercise saying that a continuous bijection from a compact space to a Hausdorff space is an isomorphism. But as you say, “surjective” will suffice here. Thanks for the correction.

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