Let (X, d) be a metric space. If A C X and c z 0, let U(A, E) be the E-

Let (X, d) be a  metric  space.  If A C X and c z 0, let U(A, E) be the E-

neighborhood of A. Let 3t' be the collcction of all (nonempty) closed, bounded

subsets of X. If A, B E X, define

D(A, B) = inf(~  1  A C U(B, E) and B C U(A, 6)).

846  Puinl wise and Cumpact Convergence  UII

(a)  Show that D is a metric on 3t'; it is called the Hausdorff metric.

(b)  Show that if (X, d) is complctc, so is (X,  D). [Hint:  Let A, be a Cauchy

sequence in 3t'; by passing to a subsequence, assume D(A,, A,+1 ) < 112".

Define A to be the set of all points x that are the limits of sequences xl, xz,

. . . such that xi E Ai for each i and d(xi, xi+,) < 112'. Show A,  + A.]

(c)  Show that if (X, d) is totally bounded, so is (2,  D). [Hint:  Given E, choose

8 < E and let S be a finite subset of X such  that the collection { Bn(x, 8) I

x E S) covers X. Let A be the collection of all nonempty subsets of S; show

that (Bo(A, E) I A E A) covers X.]

(d)  Theorem.  If X is compact in the mem'c d, then the space 3f' is compact in

the Hausdorff metric D.