A metric space is called doubling if there is some $C>0$ such that for any $r>0$ any ball of radius $r$ can be covered by $C$ balls of radius $r/2$. This is equivalent to having finite so-called Assouad dimension.
My question: If $(X,d)$ is a compact doubling metric space, is any metric space homeomorphic to it doubling?
If not, then what about when $(X,d)$ is a compact Riemannian manifold?