Compact set of real numbers with countably many limit points.

Question

Construct a compact set of real numbers whose limit points form a countable set.

My example: Let $E_1=\{1\}\cup \{1+1/n: n\in \mathbb{N}\},$ $E_2=\{1/2\}\cup \{1/2+1/n: n>2\},$ $E_3=\{1/3\}\cup \{1/3+1/n: n>6\},\dots,$ $E_k=\{1/k\}\cup \{1/k+1/n: n>k(k-1)\},\dots$

Let $F=\cup_{k=1}^{\infty}E_k$. It's easy to see that $F$ is bounded and $F$ is closed because $F'=\{1/j\}_{j\geqslant 1}$. So $F$ is compact set. Also $F'\sim \mathbb{N}$.

Is my example true?

Answer 1

I agree that $F$ is bounded and countable.

However $F$ is not closed. $\{0\}$ is a limit point that is not in $F$.

$F \cup \{0\}$ is a compact set.