# A non-denumerable set has non-denumerably many cluster points?

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

With that, since $X\subseteq (X-Y)\cup Y$, $\;$ Y is not (at-most-)countable.