Let S be an uncountable subset of a separable metric space (X,d). Then,can we find any such S in X which contains no dense subset D, dense in itself?

Question

This question arrived into my mind after observing the fact that many of the uncountable subsets is either dense or posseses a dense subset, dense in itself, even the Cantor set in R is perfect .

Answer 1

Since a subspace of a separable metric space is separable, we may as well assume that $S=X$. The answer is yes, any uncountable separable metric space $S$ contains an uncountable subset which is dense in itself. To see this write $S=C\cup D$ where $x\in C$ if some neighborhood of $x$ is countable, $x\in D$ if every neighborhood of $x$ is uncountable. Now it is easy to see that $C$ is countable (from the Lindelöf property of separable metric spaces); it follows that $D$ is uncountable, and every neighborhood of a point $x\in D$ contains uncountably many points of $D$.