## Vietoris topology and Hausdorff metric

From Kechris:

Let $X$ be a topological space.  We denote by $K(X)$ the space of all compact subsets of $X$ equipped with the Vietoris topology.

Let $(X,d)$ be a metric space with $d \le 1$.  We define the Hausdorff metric on $K(X)$, $d_H$, as follows:

$d_H(K,L) = 0$, if $K = L = \emptyset$,

$= 1$, if exactly one of $K, L$ is $\emptyset$,

$= \max \{ \delta(K,L), \delta(L,K) \}$, if $K, L \neq \emptyset$,

where $\delta (K, L) = \max_{ x \in K } d(x, L)$.