Prethick subsets and partitions of metric space

A subset $A$ of a metric space $(X,d)$ is called thick if, for every $r>0$, there is $ain A$ such that $B_{d}(a,r)subseteq A,$ where $B_{d}(a,r)={xin Xcolon d(x,a)leq r}$. We show that if $(X, d)$ is unbounded and has no asymptotically isolated balls then, for each $r>0$, there exists a partition $X=X_{1}cup X_{2}$ such that $B_{d}(X_{1},r)$ and $B_{d}(X_{2},r)$ are not thick.