Reference no: EM13831345
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.
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