Example verification: $E$ and $E'$ might not have the same limit points

Question

Let $E'$ be the set of all limit points of a set $E$.

Do $E$ and $E'$ always have the same limit points?

Proof: $E''\subset E'$ because $E'$ is closed.

But inclusion $E'\subset E''$ is false. We can take $\mathbb{R}^1$ with metric $\rho(x,y)=|x-y|$ and set $E=\{1/n: n\in \mathbb{N}\}$. Then $E'=\{0\}$ but $E''=\varnothing$.

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Is my example true?

Answer 1

Your counterexample is about as good as it gets.

Interestingly, one can investigate the behaviour of iterating the operation $'$. A key phrase in this direction is Cantor-Bendixson rank.