The rational world and the rational Urysohn space

The set Q of rational numbers is obviously an interesting topological space. In 1920, Waclaw Sierpiński gave a lovely characterisation of it. The simplest way to state it is to say that a countable, metrisable, space without isolated points is homeomorphic to Q. (Sierpiński also gave a purely topological characterisation: a countable, second countable, 0-dimensional, T₁ topological space without isolated points is homeomorphic to Q.)

In 1985, Peter Neumann named this topological space the rational world; he showed how Sierpiński’s theorem could be used to determine all the cycle types of auto-homeomorphisms of Q. (The word “auto-homeomorphism” is a mouthful, so I will simply say “automorphism” from now on.) The basic trick is very simple. If we can construct a countable metrisable space without isolated points and having an automorphism with some given cycle structure, then that is an allowable structure for an automorphism of the rational world.

Here is an example, from his paper. Consider the set of complex numbers with modulus 1 having the form exp(ikπ√2), for integers k, with the subspace topology induced from the complex numbers. It is easy to show that this set satisfies Sierpiński’s condition, and the map replacing k by k+1 is an automorphism. So the rational world has a cyclic automorphism.

In the paper, Neumann gives necessary and sufficient conditions for the cycle type of a permutation of a countable set to be realised by an automorphism of the rational world.

Let us turn this principle on its head.

In 2006, Anatoly Vershik and I looked at the Urysohn metric space and studied its isometries. This is best done via the rational Urysohn space, the Fraïssé limit of the class of finite rational metric spaces (that is, metric spaces with all distances rational). We showed that the rational Urysohn space has cyclic isometries, and hence the “real” Urysohn space has isometries all of whose orbits are dense. Now the rational Urysohn space is easily seen to be an exemplar of the rational world, and so a cyclic isometry of the rational Urysohn space gives us a cyclic automorphism of the rational world, as in Neumann’s result.

So my question is: Which cycle structures of automorphisms of the rational world are induced by isometries of the rational Urysohn space? And how do conjugacy classes of automorphisms split into conjugacy classes of isometries? It may be that there are interesting results to be found here.