Set of limit point is a closed set. [duplicate]

Let $L$ be the set of limit points of set $A$. How do you show that the set $L$ is closed? Or, is this statement not necessarily always true?

• What does $A$ lie inside? A metric space, or any topological space? – Hayden Aug 8 '14 at 11:30
• You show that if $x$ is a limit point pf $L$ then it is also a limit point of $A$. – MJD Aug 8 '14 at 11:41

The following proof works if singletons are closed.

Let $x\notin L$.

Then an open set $U$ exists such that $\left(U\backslash\left\{ x\right\} \right)\cap A=\emptyset$.

Then $U\backslash\left\{ x\right\}$ is open and from $\left(U\backslash\left\{ x\right\} \right)\cap A=\emptyset$ it follows that $\left(U\backslash\left\{ x\right\} \right)\cap L=\emptyset$.

(This because for any $y\in U\backslash\left\{ x\right\}$ we have open set $U\backslash\left\{ x\right\}$ with $\left(U\backslash\left\{ x\right\} \right)\cap A=\emptyset$, showing that $y\notin L$)

Combined with $x\notin L$ we now have $U\cap L=\emptyset$, so for $x\notin L$ we have found open set $U$ with $x\in U$ and $U\cap L=\emptyset$.

This can be done for any $x\notin L$ so allows the conclusion that $L$ is closed.