# 'set of limit points of $E$' and $E$ have the same limit points?

Let $X$ be a metric space and $E\subset X$. Let $E'$ be the set of limit points of $E$. I know that $\overline E$ and $E$ have the same limit points l, but what about $E'$?

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What about $E=\{0\}\cup\{1/n; n=1,2,3,\dots\}$ in $\mathbb R$? –  Martin Sleziak Sep 13 '12 at 18:48
As a side note: This very question and the investigation of limit points of limit points of ... as needed in Fourier ananysis was what inspired Cantor to come up with set theory. –  Hagen von Eitzen Sep 13 '12 at 19:47