In the paper by Nazarov, Treil and Volberg: Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on non-homogeneous spaces 1998, Int. Math. Res. Not. (DOI), on the page 6, they stated a version of Vitali covering theorem for a separable metric space $X$ endowed with a measure $\mu$ “not necessarily having the doubling property”:
Claim: Let $E \subset X$ be any set and $(B(x,r_x))_{x\in E}$ be a family of balls with uniformly bounded radii. Then there always exists a “countable” subfamily of disjoint balls whose triple extensions cover $E$.
The fact “countable subfamily” is strange to me. The general covering theorem for metric spaces seems to claim the existence of a subfamily without mentioning its countability. If the space is equiped with a doubling measure then it’s well known that such a countable subfamily exists. Where can I find a reference for this claim of their paper? Do the main ingredient of this countable property come from the hypothesis of “separable” metric spaces?