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Let X be a separable metric space, possibly assumed to be complete, and $B_i, i \in J$ an infinite collection of open balls. Is it true that there always exists a countable subset K of J such that the intersection of the $B_i, i \in J$ is equal to the intersection of the $B_i, i \in K$ ? If not, what kind of additional condition would be needed on X ? (the spaces X in which I am interested are not locally compact.)

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I don't think so. Here is a counterexample on the circle ${\bf R}/{\bf Z}$. The subset ${\bf Q}/{\bf Z}$ is an (uncountable) intersection of open balls: $${{\bf Q}/ {\bf Z}}= \bigcap_{x\notin {\bf Q}/{\bf Z}} \{x\}^c$$ It is not a $G_\delta$-set (a countable intersection of open sets) because of the Baire theorem.

As a side-note, all subsets of the circle are intersections of balls, but not all subsets of the circle are $G_\delta$ sets.

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