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).

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This entry was posted in Analysis, Definition, Descriptive Set Theory, Kechris, Topology. Bookmark the permalink.

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