A closed subset of the Cantor set with the usual metric is necessarily compact and totally disconnected. However, it is not necessarily perfect—for example, one can always pick a set of two points at either end of the Cantor set which will be isolated from each other.
However, if the closed subset of the Cantor set we choose is uncountable, does that necessarily imply that the closed subset is perfect?
COMMENT: The Cantor-Bendixson theorem implies that any closed subset of the Cantor set (imbued with the usual metric) can be described as the union of a Cantor set (therefore perfect and uncountable) and a countable set. However, I'm not sure if it implies that any uncountable closed subset of the Cantor set is perfect (and by extension, Cantor).