Math 524
Homework due 10/11/00
Problem 1. Let (X, d) be a compact metric space. We denote by C(X) the family of closed
subsets of X. For A, B ∈ C(X) define the Hausdorff distance between Aand Bby,
D[A, B] = sup
a∈A
d(a, B) + sup
b∈B
d(b, A),
where d(a, B) = inf{d(a, b) : b∈B}denotes the distance from the point ato the set B, and
d(b, A) = inf{d(b, a) : a∈A}the distance of the point bto the set A.
1.1 Show that Ddefines a metric on C(X).
1.2 Prove that (C(X), D) is a complete metric space.
1.3 Prove that (C(X), D) is totally bounded; that is, for every > 0 there exists a finite
cover of C(X) by -balls for the metric D.
Together, these imply that (C(X), D) is a compact metric space.
Let (X, ρ) be a metric space. Let E⊂X.
•A point x∈Xis a limit point or an accumulation point of Eif ∀r > 0, E∩B(x, r)\{x} 6=∅.
•Eis said to be perfect if Eis closed and if every point of Eis an accumulation point of E.
Problem 2. Prove that a nonempty perfect set in Rnis uncountable.
Problems from Royden: Chapter 7, Section 4: problems 14, 15.
Chapter 7, Section 5: problem 21.
Chapter 7, Section 7: problem 27.
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