Let $X$ be an infinite set, and let $G$ be the symmetric group on $X$. I want to understand $G$ by putting a topology on it, without imposing any more structure on $X$. What 'interesting' possibilities are there and what is known about them?
In particular, I have heard of the pointwise convergence topology (an open neighbourhood of g is a set of permutations that agree on some specified finite set of points), and found some papers on this, but are there any other topologies that have been studied?
What if I take the coarsest topology compatible with the group operations such that either a) the stabiliser of any subset is closed, b) the stabiliser of any partition is closed, or c) both are closed? I think these will be coarser than the pointwise convergence topology, because in the pointwise convergence topology the stabiliser of any first-order structure is closed. Is there a useful characterisation of the open subgroups and/or closed subgroups?
Best Answer
It is known that there are exactly two separable group topologies on $S_\infty$ (i.e., on the group of permutations of a countable set): one is antidiscrete and the other one is topology of pointwise convergence. This is a statement of Theorem 6.26 in here. Hence Polish topology on $S_\infty$ is unique.
There are some abstract characterizations of closed subgroups, for example they are exactly the Polish groups with a countable basis of identity consisting of open subgroups. Another characterization is: a Polish group G is homeomorphic to a closed subgroup of $S_\infty$ if and only if it admits a compatible left invariant ultrametric. You may want to look at Section 1.5 of Becker and Kechris "The Descriptive Set Theory of Polish Group Action".