Limit Point

Let $(X,d)$ be a metric space. Given $A\subseteq X$. Say $x\in X$ is a limit point of $A$ iff $B^{\ast }(x,\epsilon )\cap A\ne \varnothing \forall \epsilon >0$ Where $B^{\ast }(x,\epsilon )$ is the punctured open ball of radius $\epsilon$ centered at $x$.