Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}^c(M,\omega)$ denote the group of compactly supported Hamiltonian diffeomorphisms of $(M,\omega)$. Is the Hofer topology on $\operatorname{Ham}^c(M,\omega)$ second countable?


By the Hofer topology I mean the topology induced by the Hofer metric.



Is the Hofer topology second countable?
Since it is a metric space, your question is equivalent to asking whether it is separable (i.e. has a dense countable subset). Now, under some assumption on the manifold (compact is ok for sure, but second-countable is probably enough too), $\mathrm{Ham}\_c(M,\omega)$ is second-countable (hence separable) for the $C^1$-topology, as a subspace of the second-countable space $C^1(M,M)$. Since $C^1$ convergence implies Hofer convergence, $\mathrm{Ham}\_c(M,\omega)$ is separable for the Hofer topology.


