Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.: $$ \lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\left\{\,\sup_{x \in K_n} d(x,K),\, \sup_{y \in K} d(K_n,y) \,\right\} = 0 , $$ then what can be said of the convergence of $I_{K_n}$ to $I_K$?
What I know: Indeed, since Hausdorff convergence strictly implies convergence of the upper Kuratowski limits then, by Convergence in compact-open topology on the Sierpiński space, we know that $I_{K_n}$ converges to $I_K$ in $C(X,\{0,1\})$ for the compact-open topology if $\{0,1\}$ has the Sierpiński topology.
What I expect: Since the Hausdorff topology is strictly finer than the Sierpiński topology then, is there a topology $\tau$ on $C(X,\{0,1\})$ for which: $$ I_{X-K_n} \overset{\tau}{\to} I_{X-K} \Leftrightarrow K_n \overset{d_H}{\to} K. $$
