I want to show that the next groups are polish topological groups, which criteria should I use here?
And also which are locally compact (same question)?
The groups are:
- The group of permutations on $\mathbb{N}$, $S(\mathbb{N})$;
- The group of unitary operators on a separable Hilbert space, with the strong operator topology;
- The group of homeomorphisms on Cantor comb set with the topology of uniform convergence;
- The group of automorphisms with the usual ordering of $\mathbb{Q}$, with pointwise convergence topology;
- The group of invertible measure preserving transformations of an atomless standard probability space, $(X,M,\mu)$, with the weak topology that makes the map: $S\rightarrow \mu(S(E))$, $E\in M$ from this group to the real line continuous.
Or if you have some references that deal with these groups and this kind of questions, it will be best.
Thanks.