  We show that connected separable locally compact groups are infinitesimally
finitely generated, meaning that there is an integer $n$ such that every
neighborhood of the identity contains $n$ elements generating a dense subgroup.
We generalize a theorem of Schreier and Ulam by showing that any separable
connected compact group is infinitesimally $2$-generated.
  Inspired by a result of Kechris, we introduce the notion of a quasi
non-archimedean group. We observe that full groups are quasi non-archimedean,
and that every continuous homomorphism from an infinitesimally finitely
generated group into a quasi non-archimedean group is trivial. We prove that a
locally compact group is quasi non-archimedean if and only if it is totally
disconnected, and provide various examples which show that the picture is much
richer for Polish groups. In particular, we get an example of a Polish group
which is infinitesimally $1$-generated but totally disconnected, strengthening
Stevens' negative answer to Problem 160 from the Scottish book.
