Prethick subsets and partitions of metric spaces

A subset $A$ of a metric space $(X,d)$ is called thick if, forevery $r>0$, there is $ain A$ such that $B_{d}(a,r)subseteqA,$ where $B_{d}(a,r)={xin Xcolon d(x,a)leq r}$. We showthat if $(X, d)$ is unbounded and has no asymptoticallyisolated 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.