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Let $K(X)$ the space of all non-empty compact subsets of $X$ equipped with the topology from the Hausdorff metric.(subbasic opens: $\{K\in K(X):K\subseteq U\}$ and $\{K\in K(X):K\cap U\neq \emptyset\}$ for $U\subseteq X$ open).

A topological space $X$ is zero-dimensional if it is Hausdorff and has a basis consisting of clopens sets.

If $X$ is zero-dimensional then $K(X)$ is zero-dimensional.

As $X$ is zero-dimensional , $X$ is Hausdorff then $ K(X)$ is Hausdorff. Lack prove that $K(X)$ has a basis consisting of clopens sets.

A suggestion to show that $K(X)$ has a basis consisting of clopens sets. Thanks

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HINT: If $U$ is an open set in $X$, let

$$U^+=\{K\in\mathscr{K}(X):K\cap U\ne\varnothing\}|\;,$$ and let

$$U^-=\{K\in\mathscr{K}(X):K\subseteq U\}\;.$$

Show that if $U$ is clopen in $X$, then $U^+$ and $U^-$ are clopen in $\mathscr{K}(X)$, and note that the family of finite intersections of these subbasic open sets is a base for the topology on $\mathscr{K}(X)$.

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