Homework 5 Math 501 Due October 3, 2014 Exercise 1 Let M = Q. Show that the set S = Q ∩ [0, 1] is closed and bounded but not compact. [Hint: Consider the set A = {x ∈ S : x2 < 1/2}. Argue that this set is bounded above, construct a sequence that converges to its supremum, and go from there.] Exercise 2 Let a < b be real numbers. Prove that [a, b) is not compact. Deduce that there cannot exist a homeomorphism between [a, b] and [a, b) . Exercise 3 The set of all bounded, real-valued sequences is often denoted `∞ = {(xn ) ⊂ R : sup |xn | < ∞}. n≥1 It is not hard to show that the map d : `∞ × `∞ → R d(x, y) = sup |xn − yn | n≥1 gives a metric on `∞ , so we may regard `∞ as a metric space. In this exercise, we show that the closed unit ball B 1 (0) = {x ∈ `∞ : d(x, 0) ≤ 1} in `∞ is not compact. This, like Exercise 1, is in stark contrast to the situation in Rn , where by the Heine-Borel theorem, the closed unit ball is in fact compact. Define for each n ≥ 1 the sequence en = (0, . . . , 0, 1, 0, . . .) which has 1 in coordinate n and 0 in all other coordinates. It is obvious that each en ∈ `∞ lies in the closed unit ball B 1 (0) of `∞ . Show that the sequence 1 (en ) has no convergent subsequence. [Hint: First prove that if a sequence in a metric space converges, then it is Cauchy. What is d(en , em ) for any n 6= m?] Exercise 4 Let f : M → N be a homeomorphism and let A ⊂ M . Prove that (a) f (int A) = int f (A), (b) f (A) = f (A), (c) f (∂A) = ∂f (A). Exercise 5 Consider the following subsets of R2 : A = {x ∈ R2 : 1 ≤ |x| ≤ 2}, D = {x ∈ R2 : |x| ≤ 1}, S = {x = (x1 , x2 ) ∈ R2 : |x2 | ≤ 1}, H = {x = (x1 , x2 ) ∈ R2 : x2 ≥ 0}. Prove that (a) A and S are not homeomorphic, (b) A and D are not homeomorphic, (c) S and H are not homeomorphic. [In fact no pair of these sets is homeomorphic.] Exercise 6 Let A, B, C be sets with A ⊂ B and let f : A → C and g : B → C be functions. We say that f extends to g (or that g extends f ) if f (a) = g(a) for all a ∈ A. Let M be a metric space, let S ⊂ M be a subset, and let f : S → R be a uniformly continuous function. Let p ∈ S. (a) Suppose that (pn ) ⊂ S is a sequence that converges to p. Prove that f (pn ) converges in R. [Hint: prove that the sequence f (pn ) is Cauchy.] 2 (b) Suppose that (qn ) ⊂ S is some other sequence that converges to p. Prove that f (qn ) converges to the same point to which f (pn ) from part (a) converges. (c) Define a function f : S → R by f (x) = f (x) for x ∈ S and for p ∈ S \ S define f (p) = lim f (pn ) n→∞ where (pn ) ⊂ S is any sequence that converges to p ∈ S. Parts (a) and (b) above show that f is well-defined; that is, f (p) does not depend on the choice of sequence. Prove that f is uniformly continuous. (d) Prove that f is the unique continuous function defined on S that extends f. 3