Let $(X,d)$ be a metric space. Suppose that $\{A^n\}_{n \in \mathbb{N}}$ is a sequence of closed, non-empty subsets of $X$.
Is there a Hausdorff topology on the space of closed subsets of $X$, guaranteeing that if $A^n$ converges in this space to a $A\subseteq X$, then for any continuous function $f:X \rightarrow \mathbb{R}$, we have that $$ \sup_{x \in A}f(x)\leq \sup_{n \in \mathbb{N}}\sup_{x \in A^n}f(x) ? $$