Let $X$ and $Y$ be metrizable spaces, $Y$ compact, and $C\subseteq X\times Y$ closed.

For each $y\in Y$, let $C^y=\{x\in X:(x,y)\in C\}$, the $y$-slice of $C$. Since $Y$ is compact, the projection from $X\times Y$ onto $X$ is a closed map, and it follows that each $C^y$ is closed in $X$.

Here is my question: Suppose that $(y_n)$ is a sequence in $Y$ converging to some $y\in Y$. Is there a "reasonable" topology on the hyperspace of closed subsets of $X$ such that $(C^{y_n})$ converges to $C^y$?

"Reasonable" is up to interpretation here, but I would like it to be metrizable and closely related to the topology on $X$.

If it helps, you can take $X$ and $Y$ to be complete separable metric spaces and $C$ clopen. However, for the example I have in mind, $X$ is very much not compact (e.g., $X=\mathbb{N}^\mathbb{N}$).


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