Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the discrete topology?
If the answer is yes, does it also follow if $Homeo(M)$ the group of homeomorphisms of $M$ acts transitively on $M$ rather than the group of isometries?
I've tried thinking about this for several metric space If M is not countable, the real line serves as a counterexample, if M is not complete the rationals serve as a counterexample, and if M is not symmetric the one point compactification of a countable discrete set gives a counterexample.
Suppose that $X$ is a complete, countable, homogeneous metric space. Clearly $X=\bigcup_{x\in X}\{x\}$ is a countable union of closed sets. Every complete metric space is a Baire space, so the sets $\{x\}$ cannot all be nowhere dense. Thus, at least one of them must be isolated, and since $X$ is homogeneous, they are all isolated, and $X$ is discrete.