Here is the question
Prove that the set of limit points of a set is closed.
I'm not even understand the question fully. Is question meaning $\{E= {N_\epsilon(p) }\}$? if you said $p$ is the limit point?
How do you approach this question?
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Kingkong
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1Are we in a metric space here or a general topological space? – ncmathsadist Dec 07 '13 at 21:44
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general topological space – Kingkong Dec 07 '13 at 21:49
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Let $A$ be a subset of a topological space; denote its set of limit points by $A'$. Choose any $x$ in the closure of $A'$. Let $U$ be a neighborhood of $x$. Then $U - \{x\}$ meets $A'$.
Now pick $y\in U$ with $y\not=x$ so $y\in A'$. Then $U$ is a neigbhorhood of $y$ and $y\in A'$ so $U$ must meet $A$ somewhere other than $y$. Thus every deleted neighborhood of $x$ meets $A$. Hence $x\in A'$.
ncmathsadist
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You should prove the following: If $x_n$, $n\in\mathbb N$, are limit points of $X$ and $x_n\to x$, then $x\in X$.
Yiorgos S. Smyrlis
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That is if $x$ is in the closure, then there must be a sequence converging to $x$. This is not true. – LASV Dec 07 '13 at 22:23