Cantor sets as generalized inverse limits

We characterize when the inverse limit of a single set-valued function $F$ yields a Cantor set as its inverse limit. We do this by focusing on a subset of the domain we call ${\rm D}(F)$. When ${\rm D}(F)$ is finite, we are able to apply known results for shifts of finite type to obtain our results. We then adapt those concepts to an infinite, compact alphabet. We give general characterizations when ${\rm D}(F)$ is countable and when ${\rm D}(F)$ is uncountable. We include many examples illustrating these results.