Let $X$ be a metric space and $G$ be its group of isometries.
1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which the action is transitive?
2) It is known that if $X$ is (locally-)compact then $G$ is a (locally-)compact topological group (with the compact-open topology). The proof that I know is in volume 1 of Kobayashi & Nomizu and consists of 6 lemmae, 1 theorem and 3 corollaries, stretching on 4 pages. Does anyone know of a shorter proof?