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I can´t prove this fact in $\mathbb{R}$. I want to know how general this result is. (What topological properties are needed to prove it?) Let $X$ be a non-denumerable subset of the real numbers. How can I prove that the set of limit points of $X$ is also non-denumerable?

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I believe the common term is non-denumerable. – robjohn Aug 23 '11 at 23:58
up vote 5 down vote accepted

Let Y be the set of limit points of X.
Since X-Y does not contain any of its limit points, X-Y is discrete.
We need that all discrete subsets are (at-most-)countable.
With that, since $X\subseteq (X-Y)\cup Y$, $\; $ Y is not (at-most-)countable.

A discrete subset of a separable metric space is always (at-most-)countable.

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Beautiful, and thank! – Daniel Aug 24 '11 at 0:19

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