In the article When the collection of ε-balls is locally finite, in the second paragraph of Introduction, it is said that:
Each separable metric space $X$ admits a metric $d$ such that, the collection $\{B(x,\epsilon):x\in X\}$ of all $\epsilon$-balls is finite for every $\epsilon>0$.
I don't understand how this collection is finite if the number of elements in $X$ is infinite. Can somebody explain it?
Sets do not have duplicate elements so that $B_d(x,\varepsilon)$ could be the same set as $B_d(y, \varepsilon)$ for many pairs $x \neq y$ and then this counts as just one set in $\mathcal{B}_d(\varepsilon)$.
And eventually the result is about the collection $\{B_d(x, \varepsilon): x \in X\}$ being locally finite, which is much weaker than finite (but implies finite in a compact space). It means that every point has a neighbourhood that only intersects finitely many different elements of that set of balls. But this condition is equivalent to many stronger and weaker variants of that same condition (star-finite etc.).
It's a nice paper BTW at first glance; I didn't know it yet. It turns out that quite a few metric spaces admit such a metric.