Kechris, p. 25

(B) Exercise (4.21).

Show that the Hausdorff metric is compatible with the Vietoris topology.

Proof.

Let be a metric space with . Let denote the topology from the Hausdorff metric on , with basis , denote the Vietoris topology on , with basis

.

If , .

(i) .

Proof.

Let . Suppose . Then, since and , we’re done.

So, suppose now that . Cover with finitely many balls , , of radius , such that, for all , . Let .

I will show that satisfies .

Let . That is , and for all , . To see that , I check that and . Let . Then , so for some , so . Similarly, if , .

So .

(ii) .

Proof.

Let , . If , then . So suppose that . I have that , and , .

Claim 1: .

Proof.

Suppose . Then, I can pick such that . Let such that . Then, has a convergent subsequence, . But then , which is a contradiction since is closed.

Let . Let . Then such that . Then let .

Claim 2: .

Proof.

Let . That is, . Then, , . Then, for each , . So there exists such that . Hence , and .

Finally, let . Then , so (see Claim 1). So , and .

So .

And hence .