When the collection of ε-balls is locally finite

Consider the class N of metrizable spaces which admit a metric d such that, for every ε>0, the collection {B(x,ε): x∈X} of all ε-balls is locally finite. We show that N is precisely the class of strongly metrizable spaces, i.e., X∈N iff X is homeomorphic to a subspace of κω×[0,1]ω for some cardinal κ (where κ carries the discrete topology). In particular, this shows that not every metrizable space admits such a metric, thereby answering a question of Nagata.