Let $(X,d)$ be a metric space, let $A\subseteq X$ be a subset of $X$, and let $(A_n)_{n=1}^{\infty}$ be a sequence of subsets of $X$ such that
$$ \lim_{n\to\infty}d(x,A_n)=d(x,A), \quad \forall x\in X \tag{1}$$
where $d(x,Y)=\inf_{y\in Y}d(x,y)$.
In what sense does $(A_n)_{n=1}^{\infty}$ converge to $A$? This is weaker than usual set convergence such as $\bigcap_{n=1}^{\infty}A_n=A$. For example, under definition $(1)$, the sequence of sets $\{1\},\{1/2\},\{1/3\},\dots$ converges to the set $\{0\}$. Is there a name for this type of convergence?