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

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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 $2^{\aleph_0}$.

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