ANALYSIS WORKSHEET II: METRIC SPACES

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1 ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair x, y of points in X a non-negative real number d(x, y) 0. Furthermore, the metric must satisfy the following four axioms: (a) For any x X, we have d(x, x) = 0. (b) (positivity) For any distinct x, y X, we have d(x, y) > 0. (c) (symmetry) For any x, y X, we have d(x, y) = d(y, x). (d) (triangle inequality) For any x, y, z X, we have d(x, z) d(x, y) + d(y, z). S16#6. A metric ρ in a metric space (X, ρ) is said to be an ultrametric if x, y, z X : ρ(x, y) max{ρ(x, z), ρ(y, z)}. Prove that, in this metric, every open ball {y : ρ(x, y) < r} is closed and every closed ball {y : ρ(x, y) r} is open. Definition 2. Let (X, d) be a metric space, let Y be a subset of X, and let E be a subset of Y. We say that E is relatively open with respect to Y if it is open in the metric subspace (Y, d Y Y ). Similarly, we say that E is relatively closed with respect to Y if it is closed in the metric space (Y, d Y Y ). Exercise 1. Let (X, d) be a metric space, let Y be a subset of X, and let E be a subset of Y. (a) E is relatively open with respect to Y iff E = V Y for some set V X which is open in X. (b) E is relatively closed with respect to Y iff E = K Y for some set K X which is closed in X. Definition 3. Let (X, d) be a metric space. We say that X is disconnected iff there exist disjoint non-empty open sets V and W in X such that V W = X. We say that X is connected iff it is non-empty and not disconnected. Definition 4. Let (X, d) be a metric space. Two subsets A and B of X are said to be separated if both A B and A B are empty. 1

2 2 ANALYSIS WORKSHEET II: METRIC SPACES Exercise 2. Let X be a metric space. (i) X is disconnected iff there exist disjoint non-empty closed sets V and W in X such that V W = X. (ii) X is disconnected iff there exist non-empty separated sets V and W in X such that V W = X. (iii) X is disconnected iff X contains a non-empty proper subset which is simultaneously open and closed. Definition 5. Let (X, d) be a metric space, and let Y be a subset of X. We say that Y is connected iff the metric space (Y, d Y Y ) is connected. F14#2. Let A, B be two closed subsets of R n such that A B and A B are connected. Prove that A is connected. Definition 6. Let (X, d) be a metric space, and let E be a subset of X. We say that E is path-connected iff, for every x, y E, there exists a continuous function γ : [0, 1] E such that γ(0) = x and γ(1) = y. Exercise 3. Every path connected set is connected. Exercise 4. Every open connected subset of R n is path-connected. In the last exercise, R n is assumed equipped with its standard metric. Remember, that the statement connected implies path connected is not true in general. Exercise 5. Let (X, d) be a metric space and E a subset of X. The following are equivalent (a) E is complete and totally bounded. (b) (Bolzano-Weierstrass) Every sequence in E has a subsequence that converges to a point of E. (c) (Heine-Borel) If {V λ } λ Λ is a cover of E by open sets, there is a finite set A Λ such that {V λ } λ A covers E. Definition 7. Let (X, d) be a metric space and E a subset of X. If E satisfies any one (hence all) of the properties (a)-(c) of the previous exercise then E is called compact. S18#11. Let (X, ρ) be a compact metric space and let f : X X be an isometry (meaning that ρ(f(x), f(y)) = ρ(x, y) for all x, y X). Prove that f is onto; that is, f(x) = X. Hint: Consider x / f(x) and follow the iterates of f on x.

3 ANALYSIS WORKSHEET II: METRIC SPACES 3 Definition 8. Let X be a metric space. A chain of sets is a finite series of sets E 1, E 2,..., E n such that E i meets E i+1, i = 1, 2,..., k 1. Exercise 6. Let X be a metric space. Then X is connected iff given any open cover, or finite closed cover, {E α } of X, any two members of {E α } are the first and last members of a chain of sets in {E α }. Exercise 7. Let X be a compact metric space and E X closed. compact. Then E is S17#10. Let K R n be compact. Suppose that for every ɛ > 0 and every pair a, b K, there is an integer n 1 and a sequence of points x 0,..., x n K so that x 0 = a, x n = b, and x k x k 1 < ɛ for every 1 k n. (i) Show that K is connected. (ii) Show by example that K may not be path connected. Exercise 8. Let K R n be compact and connected. Then K is ɛ-chainable as in S17#10. Definition 9. A metric space (X, d) is said to be complete iff every Cauchy sequence in (X, d) is in fact convergent in (X, d). Exercise 9. Let X be a compact metric space and C(X) the space of continuous real-valued functions on X endowed with the supremum norm. Show that C(X) with the norm distance is a complete metric space. Exercise 10. Let (X, d) be a complete metric space and E X a closed subset. Then (E, d E E ) is complete. Exercise 11. Show that ρ(x, y) = tan(y) tan(x) defines a complete metric on ( π/2, π/2).

4 4 ANALYSIS WORKSHEET II: METRIC SPACES F15#6. Let X := R \ {0}. Find a metric ρ on X with the following properties: (1) (X, ρ) is a complete metric space, and (2) if {x n } n=1 X and x X, then lim x n x = 0 x n x in (X, ρ). n Prove both properties, as well as all of your other assertions, in full detail. Definition 10. Let (X, d) be a metric space. A subset E X is perfect if E is closed and if every point of E is a limit point of E. Exercise 12. Let P be a nonempty perfect set in R n. Then P is uncountable. S14#12. Assume [0, 1] = n=1 I n where I n = [a n, b n ] and I n I m = whenever n m. (a) Let E = {a n : n 1} {b n : n 1} be the set of endpoints of the intervals above. Prove E is closed. (b) Prove no such family of intervals {I n } can exist. The exercise preceding S14#12 and the strategy it suggests for that problem are not unrelated to the following exercise, a version of the Baire Category Theorem. Exercise 13. If R n = n=1 F n, where each F n is a closed subset of R n, then at least one F n has non-empty interior. A stronger statement, a theorem of Sierpiński is stated as the next exercise (which you can safely skip). Definition 11. A topological space X is Hausdorff if for all x, y X, x y, there exist disjoint open sets U, V X with x U and y V. A continuum is a compact connected Hausdorff space. Exercise 14. If a continuum X has a countable cover {X j } j=1 closed subsets, then at most one of the sets X j is non-empty. by pairwise disjoint A special case of the preceding statement that is still stronger than S14#9 is that the unit interval [0, 1] cannot be written as the countable union of pairwise disjoint non-empty closed sets. On mathoverflow.net Tim Gowers suggests a nice method of proof based on the following exercise.

5 ANALYSIS WORKSHEET II: METRIC SPACES 5 Exercise 15. A nested sequence of open intervals has non-empty intersection provided neither end point is eventually constant. Definition 12. A metric space is called separable if it contains a countable dense subset. Definition 13. Let X be a metric space. A collection {V α } of open subsets of X is said to be a base for X if the following is true: For every x X and every open set U X such that x U, we have x V α U for some α. Exercise 16. Let X be a separable metric space. Then X has a countable base. Exercise 17. Let X be a compact metric space. Then X is separable. S17#9. Let (X, dist) be a bounded metric space and let C(X) denote the space of bounded continuous real-valued functions on X endowed with the supremum norm. Suppose C(X) is separable. (i) Show that for every ɛ > 0, there is a countable set Z ɛ X so that x X, z Z ɛ, such that y X, dist(x, y) dist(z, y) < ɛ. (ii) Deduce that X is separable. Exercise 18. Let l be the set of real bounded sequences (a i ) with the distance Then l is not separable. d ((a i ), (b i )) = sup a i b i. i N F16#11. We define a metric space (X, dist) as follows: X := {f : [0, 1] [0, 1] f is continuous and f(1) = 0} dist(f, g) = inf{r [0, 1] f(t) = g(t) for all r t 1} Prove any TWO of the following statements about (X, dist): (a) It is not compact. (b) It is not connected. (c) It is not separable. (d) It is not complete. Definition 14. Let (X, d) be a metric space and let E X. Let F be a family of (complex valued) functions defined on E. We say F is pointwise bounded on E if for all x E there exists M(x) 0 such that for all f F, f(x) M(x).

6 6 ANALYSIS WORKSHEET II: METRIC SPACES Definition 15. Let (X, d) be a metric space and let E X. Let F be a family of (complex valued) functions defined on E. We say F is uniformly bounded on E if there exists M 0 such that for all f F and for all x E, f(x) M. Definition 16. Let (X, d) be a metric space and let E X. Let F be a family of (complex valued) functions defined on E. We say F is equicontinuous on E if given ɛ > 0 and x E there exists δ = δ(x, ɛ) > 0 such that for all y E with d(x, y) < δ and for all f F, we have f(x) f(y) < ɛ. Exercise 19. Let (X, d) be a metric space and let E X be compact. Let F be an equicontinuous family of functions defined on E. Then given ɛ > 0 there exists δ = δ(ɛ) > 0 such that for all x, y E with d(x, y) < δ and for all f F, we have f(x) f(y) < ɛ. F17#12. Let X be a compact metric space and C(X) the space of continuous real-valued functions on X endowed with the supremum norm. Let F C(X) be non-empty. Prove the following version of Arzelà-Ascoli s theorem: F is compact F is closed, bounded and equicontinuous Give precise definitions of all the terms used in this equivalence. S18#12. Let F be a family of real-valued functions on a compact metric space taking values in [ 1, 1]. Prove that if F is equicontinuous, then g(x) = sup{f(x) f F} is continuous. The problem labeled S18#9, for example, refers to question 9 of the spring 2018 basic exam. Aside from the basic exam problems, the majority of the material is taken directly from Analysis I & II by Terence Tao and Principles of Mathematical Analysis by Walter Rudin.