# Convergence of slices in some topology on hyperspace of closed sets

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., $$X=\mathbb{N}^\mathbb{N}$$).