Difference between "limit point" and "points in the closure" Given a topology $(X,T)$, $A\subset X$, $x \in X$ is a limit point of A if $\forall$ open $U$ that contains $x$, $(U\cap A)$\ {$x$} $\neq \emptyset$. $x \in X$ is in $cl(A)$ if $\forall$ open $U$ that contains $x$, $U\cap A$ $\neq \emptyset$. Is there any example that a point in the the closure of $A$ is not a limit point of A? Are the two equivalent in metric space?

consider a finite set A of singletons in $\mathbb{R}$, there are no limit points, but they are al in $cl(A)$
