Let $(X, d)$ be a compact metric space. For $x \in X$ and $\epsilon > 0$, define $B_{\epsilon}(x) := \{y \in X \mid d(x, y) < \epsilon\}$. For $C \subseteq X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := \cup_{x \in C} B_{\epsilon}(x)$. Let $\mathcal{K}$ be the set of non-empty compact subsets of $X$. For $C, C^{\prime} \in \mathcal{K}$, define $\delta\left(C, C^{\prime}\right) = \inf\{\epsilon \mid C \subseteq B_{\epsilon}\left(C^{\prime}\right)$ and $C^{\prime} \subseteq B_{\epsilon}(C)\}$. Show that $(\mathcal{K}, \delta)$ is a compact metric space.
Let $(X, d)$ be a compact metric space. For $x \in X$ and $\epsilon > 0$, define $B_{\epsilon}(x) := \{y \in X \mid d(x, y) < \epsilon\}$. For $C \subseteq X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := \cup_{x \in C} B_{\epsilon}(x)$. Let $\mathcal{K}$ be the set of non-empty compact subsets of $X$. For $C, C^{\prime} \in \mathcal{K}$, define $\delta\left(C, C^{\prime}\right) = \inf\{\epsilon \mid C \subseteq B_{\epsilon}\left(C^{\prime}\right)$ and $C^{\prime} \subseteq B_{\epsilon}(C)\}$. Show that $(\mathcal{K}, \delta)$ is a compact metric space.