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 [this post][1] 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 [Sierpinki topology][2].  

**What I expect:**  Since the Hausdorff topology is strictly finer than the Sierpinski topology then, is there a topology $\tau$ on $C(X,\{0,1\})$ for which:
$$
I_{X-K_n} \overset{\tau}{\to} I_{X-K} \Leftrightarrow K_n \overset{d_H}{\to} K.
$$


  [1]: https://mathoverflow.net/questions/382810/convergence-in-compact-open-topology-on-the-sierpi%C5%84ski-space
  [2]: https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space