One of my favourite textbooks is Simmons’ Introduction to Topology and Modern Analysis. In the introduction, the author distinguishes two types of mathematics: the rare jewels, like the formula saying that the Riemann zeta function evaluated at 2 is π2/6; and the general theories, such as the book is about (Part 1 is general topology, Part 2 is linear algebra up to the spectral theorem, and Part 3 merges them to give functional analysis).
Two jewels of a kind of mathematics that aspires to be a general theory are among my favourite objects. I have known about the random graph for nearly 40 years, and can no longer remember when I first heard about it. Fifteen years ago, when I talked about it at the ECM in Barcelona, Anatoly Vershik introduced me to the Urysohn space. Now, in a lecture by Slavomir Solecki, I have met a third such object, the pseudo-arc.
If you have an eye for such things, you will have noticed that each of these has the definite article as part of its name; this implies that each such object is unique, and hints that however you try to construct it (within a wide range) you will end up with the same thing. The same is true of the pseudo-arc.
First, a brief comment about Baire category, a way of saying “almost all” which is complementary to saying “measure 1” in a probability space. In a complete metric space, a subset is residual if it contains a countable intersection of open dense sets. Residual sets are “large”: they are non-empty (this is the Baire category theorem); the intersection of countably many residual sets is residual; and a residual set meets every open set.
Now consider the set of compact connected subsets of the unit square. There is a metric, the Hausdorff metric, which makes this set into a complete metric space. (Two sets are within distance ε in this metric if the ε-neighbourhood of either set contains the other.) Now there is an element P of this space (a compact connected set) with the property that the set of elements homeomorphic to P is residual; in other words, “almost all compact connected sets” are homeomorphic to P“. This P is the pseudo-arc.
Now if we started with the unit cube in n dimensions (with n>1), or countably many dimensions, we would obtain an object whose homeomorphism class is residual; but the object we obtain is in all cases homeomorphic to P.
(Why not one dimension? There is again a unique object, but it is not P. Compact connected subsets of [0,1] are intervals and points, and almost all of them are intervals; all intervals are homeomorphic.)
Aside: How depressing that it has taken so long before I learned about this object. Lines of communication across the continuous/discrete divide of mathematics clearly don’t work too well!
Slavomir’s talk explained why people on our side of the divide should know about it. Here is an outline of his talk.
- We know very well Fraïssé’s construction. We take a class of finite structures with the amalgamation property (this says that, given A,B,C in the class with injective maps from A into each of B and C, there is a structure D in the class and injective maps from B and C into D such that the obvious diagram commutes). Fraïssé tells us that there is a unique countable limit of the class, a homogeneous structure such that all our finite structures have injective maps to it. Now turn all the arrows around and replace “injective” by “surjective”; the analogue of Fraïssé’s theorem holds, so that the inverse limit of the class is the unique dual homogeneous object which has surjections to all the finite structures in our class.
- Now perform this construction for the class of finite objects which are paths with a loop at every vertex. Homomorphisms of such objects correspond to walks which at each step can move either way or stay where they are. These things can be arbitrarily tangled! But they form a dual Fraïssé class, and so the dual homogeneous structure P exists.
- This is not what we want; an inverse limit of finite structures with the discrete topology is going to be a Cantor space, that is, totally disconnected. But there is a natural way of glueing some pairs of points together to make a connected space which turns out to be the pseudo-arc!
The last step is perhaps described by analogy. The usual Cantor space consists of points in the unit interval whose ternary expansion contains only the digits 0 and 2. If we map each point by replacing 0 and 2 by 0 and 1 and regarding this as the binary expansion, our map is one-to-one except for a small set (points whose expansion ends with an infinite string of 2s are identified with points whose expansion ends with an infinite string of 0s), and the resulting set is the whole interval.
This approach allows them to do some interesting things. They have a new proof of a theorem of Bing stating that the homeomorphism group of the pseudo-arc acts transitively on its points, and they are developing higher homogeneity properties. These results are hard work since points are quite difficult to see in inverse limits!