  In his pioneering work in the field of Inductive Inference, Gold (1967)
proved that a set containing all finite languages and at least one infinite
language over the same fixed alphabet is not learnable in the exact sense.
Within the same framework, Angluin (1980) provided a complete characterization
for the learnability of language families. Mathematically, the concept of exact
learning in that classical setting can be seen as the use of a particular type
of metric for learning in the limit. In this short research note we use
Niyogi's extended version of a theorem by Blum and Blum (1975) on the existence
of locking data sets to prove a necessary condition for learnability in the
limit of any family of languages in any given metric. This recovers Gold's
theorem as a special case. Moreover, when the language family is further
assumed to contain all finite languages, the same condition also becomes
sufficient for learnability in the limit.
