Topology generated by the subcollections of compact sets of a metric space

For a metric space $(X,d)$ let $\mathbb{K}X$ be the collection of compact subsets of $X$. Give $\mathbb{K}X$ the topology generated by the sets: $$W(U,K)=\{C\in\mathbb{K}X:(K\cup C)-(K\cap C)\subseteq U\}$$

Here $K\subseteq X$ is compact, $U\subseteq X$ is open. Define: $$D:\mathbb{K}X\times \mathbb{K}X\to\mathbb{R}, D(C,K)=\text{min}\{d(u,v):u\in C,v\in K\}.$$

How to show that $D$ is continuous?

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