Set whose subsets are all convex

Question

I am looking for a metric space in which all subsets are convex sets. I tried to find examples of such sets, but didn't find any so far. Thank you.

Answer 1

A metric space with exactly two points cannot be convex, therefore you are looking for the space of a single point.

Answer 2

First of all, to define a convex set you need two operations on your space (i.e, it has to be a vector space), since a set $C$ is convex iff $tC+(1-t)C = C$. And given a vector space with two distinct points $x\neq y$, take $A=\left \{x, y\right \}$ is not convex.

Answer 3

What about ({0,1},HD) where HD is the Hamming distance? The empty set, {0}, {1} and the segment [0,1] are all convex, right?