Is every point in $A $ a limit point of $A$. ? Yes/No

Question

Is the following statement is true/false ?

Let $(X,d)$ be a metric space. Let $A$ be closed set in $X$. Every point in $A $ is a limit point of $A$.

My attempt : I thinks this statement is true . Take $X = \mathbb{R}$ and $A =[0,1]$

Answer 1

If $A =\{0\} \cup [2,3]$ then $ 0 \in A$ but $0$ is not a limit point of $A$.