# Are non-empty perfect sets in separable metric spaces uncountable?

I know this is true for $R^{k}$, but can it be generalized to all separable metric spaces?

No. The set $\mathbb{Q}$ of rationals is a separable metric space, and is a countable perfect subset of itself.
If you ask about complete metric spaces, then this is true and follows from the Baire category theorem. Indeed, it can be shown that a perfect subset of a complete metric space has cardinality at least $2^{\aleph_0}$.