Construct a compact set of real numbers whose limit points form a countable set? (Rudin Exercise) I think $S = \{r \in \Bbb Q: 0 \leq r \leq 1\}$ in $\Bbb R$ can be a set that satisfies the conditions. First, it is compact by the Heine-Borel theorem since it is closed and bounded. It is closed because it contains all its limit points. Every rational number is a limit point; if we take an open neighbourhood of radius s, we can find some rational number contained in the open ball. Finally, the rationals are countable.

Is this correct?

Thank you.
What about $\{\frac 1n:n\in Z^+\} \cup \{0\}$?
What about $\{\frac 1n:n\in Z^+\} \cup \{0\}$?