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M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 EXTRA PROBLEM SHEET Exercise 1. Let (X, dX ) and (Y, dY ) be metric spaces. Define dp : (X ⇥ Y ) ⇥ (X ⇥ Y ) ! R p (x1 , y1 ), (x2 , y2 ) 7 ! p dX (x1 , x2 )p + dY (y1 , y2 )p A generalization of the Cauchy–Schwartz inequality, which you may assume without proof, is Hölder’s inequality. This states that, if p 2 [1, 1) and q 2 [1, 1) are such that 1/p + 1/q = 1, then: n n n ⇣X ⌘1/p ⇣ X ⌘1/q X p ai bi |ai | |bi |q i=1 i=1 i=1 for all a, b 2 R . Use this to show that dp is a metric on X ⇥ Y . n Exercise 2. Let X denote the set of all continuous functions from the interval [ 1, 1] to R. Define: d1 : X ⇥ X ! R Z 1 (f, g) 7 ! |(f (t) d1 : X ⇥ X ! R (f, g) 7 ! sup |f (t) g(t)| dt g(t)| t2[ 1,1] 1 Both (X, d1 ) and (X, d1 ) are metric spaces. (You do not need to prove this.) Consider the function ev0 : X ! R f 7 ! f (0) Is ev0 a continuous function on (X, d1 )? Is ev0 a continuous function on (X, d1 )? 1 M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 1 Exercise 1. Let p be a prime number. Define a function d : Z ⇥ Z ! R by ( 0 if m = n d(m, n) = 1 if m 6= n, where m n = pr 1 q with q 2 Z not divisible by p. r Show that d is a metric on Z. Exercise 2. Let C([a, b]) denote the set of continuous functions from [a, b] to R, and let C 1 ([a, b]) denote the set of di↵erentiable functions f : [a, b] ! R such that f 0 is continuous. Let: d1 (f, g) = sup |f (x) g(x)| x2[a,b] 1 This defines a metric on both C([a, b]) and C ([a, b]). (1) Consider the map: Int : C([a, b]), d1 ! C 1 ([a, b]), d1 Z x f 7! f (t) dt a Is Int continuous? (2) Consider the map: Diff : C 1 ([a, b]), d1 ! C([a, b]), d1 f 7! f 0 Is Diff continuous? 1 2 PROBLEM SHEET 1 Exercise 3. Let (X, d) be a metric space and A be a subset of X. Show that x 2 @A if and only if, for all ✏ > 0, we have that B✏ (x) \ A and B✏ (x) \ (X \ A) are both non-empty. Exercise 4. Let (X, d) be a metric space and A be a subset of X. For x 2 X, define d(x, A) = inf{d(x, a) : a 2 A} Show that: (1) d(x, A) = 0 if and only if x 2 A. (2) for all y 2 X, d(x, A) d(x, y) + d(y, A). (3) the map x 7! d(x, A) defines a continuous function from X to R. M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 2 Exercise 1 Let (X, d) be a metric space. (1) Show that, given any two distinct points x, y 2 X there exist open sets U , V in X with x 2 U , y 2 V and U \ V = ;. (2) Suppose that B is a bounded subset of X and that C ⇢ B. Show that C is bounded, and that diam C diam B. Exercise 2 Let f : R ⇥ R ! R be defined by: f (x, y) = ( xy x2 +y 2 0 (x, y) 6= (0, 0) (x, y) = (0, 0 Show that the restriction of f to the subset R ⇥ R \ {(0, 0)} is continuous. Is f continuous? Exercise 3 (1) Let (X, d) be a metric space and let A1 , . . . , Am be subsets of X. Show that: ✓ i=m [ ◆ i=m [ Ai = Ai i=1 i=1 (2) Let (X, d) be a metric space and let Ai , i 2 I, be subsets of X. Show that: ✓ i=m \ ◆ i=m \ Ai ✓ Ai i=1 i=1 (3) Give an example of a metric space (X, d) and subsets A, B of X such that: A \ B 6= A \ B Exercise 4 Compute the closure of each of the following sets in R: (1) [1, 1) (2) R \ Q n (3) { n+1 : n 2 N} 1 (4) { n : n 2 N, n 2} [ {0, 1, 2} 1 2 PROBLEM SHEET 2 Exercise 5 The Cantor set C is defined as follows. Let C0 = [0, 1] and, for n 0, let Cn+1 be the set obtained from Cn by taking each maximal closed interval I contained in Cn and removing from I the open interval that forms the middle third of I. Set: \ C= Cn n 0 n (1) Show that Cn is a disjoint union of 2 closed intervals; (2) Show that C is closed. (3) Show that C is non-empty. (4) Optional: show that C is uncountable. (5) Optional: show that every point in C is an accumulation point. (6) Optional: show that C has empty interior. The Cantor set is an example of an uncountable set with Lebesgue measure zero. Items (2) and (5) here show that C is a so-called perfect set. Item (6) shows that C is nowhere dense in R. M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 3 Exercise 1. Suppose that X is an infinite set equipped with the cofinite topology. (1) Let A ⇢ X be a finite set. Compute A. (2) Let A ⇢ X be an infinite set. Compute A. Exercise 2. Let X be a non-empty set and let T1 and T2 be topologies on X. Must T1 \ T2 be a topology on X? Must T1 [ T2 be a topology on X? 1 2 PROBLEM SHEET 3 Exercise 3. Let A be a subset of the topological space X. Show that @A = A \ X \ A. Exercise 4. Let X and Y be topological spaces. Let: B = {U ⇥ V : U is open in X and V is open in Y } The product topology on X ⇥ Y is T = {Z ⇢ X ⇥ Y : Z is a union of elements of B} This is a topology on X ⇥ Y . (You do not need to show this.) (1) Show that the projection maps: p1 : X ⇥ Y ! X (x, y) 7! x p2 : X ⇥ Y ! Y (x, y) 7! y are continuous, where X ⇥ Y is given the product topology. (2) Let Z be a topological space. Show that a map f : Z ! X ⇥ Y is continuous if and only if p1 f and p2 f are continuous. M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 4 Exercise 1. (1) Let X be a non-empty set equipped with the discrete topology. Show that X is compact if and only if X is finite. (2) Let X be a topological space, and let Y and Z be compact subsets of X. Show that Y [ Z is compact. Exercise 2. Let X be a compact topological space and let f : X ! Y be a continuous map of topological spaces. Show that f (X) is compact. (This would make an excellent exam question. If you can do it, then you are probably getting the hang of compactness.) 1 2 PROBLEM SHEET 4 Exercise 3. Let X be a compact topological space. Suppose that, for each n 2 N, Vn is a closed non-empty subset of X and that: Show that: V0 ◆ V1 ◆ V2 ◆ · · · \ n 0 Vn 6= ? Is this statement true without the compactness hypothesis? Exercise 4. Suppose that X is a topological space and that A, B are connected subsets of X such that A \ B 6= ?. Show that A [ B is connected. M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 5 Exercise 1. Let (X, T ) be a topological space. Let 1 be an object not in X, and set: X̌ = X [ {1} Ť = T [ V [ {1} : V ✓ X such that X \ V is compact and closed (1) Show that Ť is a topology on X̌. (2) Show that the topological space (X̌, Ť ) is compact. (3) Show that the topological space (X̌, Ť ) contains (X, T ) as a subspace. (4) Suppose that X = R with the usual topology. What is (X̌, Ť )? (5) Suppose that X = R2 with the usual topology. What is (X̌, Ť )? The space (X̌, Ť ) is called the one-point compactification of X. Exercise 2. (1) Show that the annulus {(x, y) 2 R2 : 1 < x2 + y 2 < 2} is path connected. (2) Show that the region {(x, y) 2 R2 : x < 0 or x > 1} is not path-connected. 1 2 PROBLEM SHEET 5 Exercise 3. Consider the topologist’s sine curve T , which is the subspace of R2 defined by: T = {(0, 0)} [ (x, sin x 1 ) : x 2 (0, 1) (1) Draw a picture of T . (2) Show that T is connected. (3) Show that T is not path-connected. M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 6 Exercise 1. Consider the following functions fn : [0, 1] ! R. In which case does the sequence (fn )n 1 converge uniformly on [0, 1]? x (1) fn (x) = 1+nx xn (2) fn (x) = 1+x n 2 (3) fn (x) = nx(1 x2 )n Exercise 2. Construct functions fn : R ! R such that none of the fn is continuous at 0 2 R but that (fn )n 1 converges uniformly on R to a continuous function. 1 2 PROBLEM SHEET 6 Exercise 3. Let X be the metric space [1, 1), considered as a subspace of R. Let f : X ! X be the map x 7! x + x 1 . Show that: (1) X is complete; (2) |f (x) f (y)| < |x y| for all x, y 2 X; (3) f has no fixed point. Exercise 4. We say that a metric space X is totally bounded if and only if for each ✏ > 0 there exist finitely many points x1 , . . . , xN 2 X such that: B✏ (x1 ), . . . , B✏ (xN ) is a cover of X. Show that a metric space X is compact if and only if it is complete and totally bounded. (Hint: for one direction, show that complete and totally bounded implies sequentially compact.) M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 7 Exercise 1. Give an explicit homotopy between the following paths in R2 : f : [0, 1] ! R2 g : [0, 1] ! R2 t 7 ! cos ⇡t, sin ⇡t t 7 ! cos ⇡t, sin ⇡t Exercise 2. We say that a subspace D of Rn is star-shaped if and only if there exists a point x0 2 D such that, for all x 2 D, the straight line segment from x0 to x lies entirely within D. Let D be a star-shaped subspace of Rn . (1) Show that D is path-connected. (2) Show that ⇡1 (D, x0 ) = {1}. When combined with the next exercise, this shows that D is simply-connected. 1 2 PROBLEM SHEET 7 Exercise 3. This exercise shows that, up to isomorphism, the fundamental group ⇡1 (X, x0 ) depends only on the path-component of X that contains x0 . Let X be a topological space, let x0 and x1 be points in X, and let : [0, 1] ! X be a path 1 from x0 to x1 . Let 1 : [0, 1] ! X be the reverse path: (t) := (1 t). Define a map M : ⇡1 (X, x0 ) ! ⇡1 (X, x1 ) (1) (2) (3) (4) (5) Show that M Show that M Show that M Show that if Suppose that M = M 0? [f ] 7 ! [ ⇤ f ⇤ 1 ] is well-defined. That is, show that if f ⇠ g then ⇤ f ⇤ 1 ⇠ ⇤ g ⇤ 1 . is a homomorphism of groups. is an isomorphism. (Hint: what is (M ) 1 ?) ⇠ 0 then M = M 0 . and 0 are both paths in X from x0 to x1 . Is it always the case that M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 PROBLEM SHEET 8 Exercise 1. Consider the map p : R ! S 1 given by t 7! cos 2⇡t, sin 2⇡t . Show that this is a covering map. That is, exhibit an open cover {Ui : i 2 I} of S 1 such that for each i 2 I, p 1 (Ui ) is a disjoint union: a p 1 (Ui ) = V↵ ↵ where p|V↵ gives a homeomorphism from V↵ to Ui . Exercise 2. Path lifting. Let : I ! S1 be a path in S 1 from x0 to x1 and let x̃0 2 R be such that p(x̃0 ) = x0 . We will show that there is a unique path ˜ : I ! R such that ˜ begins at x̃0 and that p ˜ = . To do this: (1) Show that there exist 0 = t0 < t1 < · · · < tN = 1 such that ([ti 1 , ti ]) lies entirely within one of the open sets Uj that you constructed in Exercise 1. (Hint: Lebesgue number!) (2) Suppose that we have constructed the lifted path ˜ on the subinterval [0, ti 1 ]. Explain how to extend this to get the lifted path ˜ on the subinterval [0, ti ]. (Hint: use the fact that p is a covering map.) (3) Show that the lifted path ˜ : I ! R thus constructed is the unique path ˜ : I ! R such that ˜ begins at x̃0 and that p ˜ = . 1 2 PROBLEM SHEET 8 Exercise 3. Homotopy lifting. Suppose that Y is a topological space, that F : Y ⇥ I ! S 1 is a continuous map, and that f˜: Y ! R is such that p f˜ = F |Y ⇥{0} . We will show that there is a unique continuous map F̃ : Y ⇥ I ! R such that F̃ |Y ⇥{0} = f˜ and that p F̃ = F . (1) Show that, for each y0 2 Y , we can find a finite open cover V1 , . . . , VM of {y0 }⇥I ⇢ Y ⇥I such that each Vj is connected and F (Vj ) lies inside one of the sets Ui constructed in Exercise 1. (2) Write Vy0 = V1 [ · · · [ VM . Arguing as in Exercise 2, show that there is a unique map F̃y0 : Vy0 ! R such that p F̃y0 = F |Vy0 and that F̃ (y0 , 0) = f (y0 ). (3) Show that Vy0 contains N ⇥ I for some open neighbourhood N of y0 in Y . Deduce that the lifts F̃y0 agree for varying y0 2 Y . This shows that defining F̃ : Y ⇥ I ! R by F̃ (y, t) = F̃y (y, t) gives the unique continuous map F̃ : Y ⇥ I ! R such that F̃ |Y ⇥{0} = f˜ and that p F̃ = F . M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 MOCK FINAL EXAM QUESTIONS Abstract. There are too many questions here – the final itself will have four questions. Also question 6 is probably too long to be a good exam question. But nonetheless these should give you some useful practice ahead of the final exam. I will post solutions on the class website after the Easter holiday. (1) Give examples of the following, or prove that no such examples exist. (a) A metric space X and open subsets Ui , i 2 I, of X such that \i2I Ui is not open. (b) An unbounded metric space X and a sequence (xn )n 1 of points of X such that (xn ) has no convergent subsequence. (c) A compact metric space X such that X is not complete. (d) A topological space X, a point x0 in X and a loop : [0, 1] ! X based at x0 such that is homotopic to , where is the reversed loop: (t) = (1 t). (e) Topological spaces X and Y and a continuous map f : X ! Y such that X is compact, Y is Hausdor↵, and f (X) is not closed. (f) Topological spaces X and Y , an open set U ⇢ X, and a continuous map f : X ! Y such that f (U ) is not open. (2) (a) Suppose that X is a topological space and that A, B are connected subsets of X such that A \ B 6= ?. Show that A [ B is connected. (b) Consider the subspace T of R2 defined by: T = {(0, 0)} [ (x, sin x 1 ) : x 2 (0, 1) Is T connected? Is T path-connected? (3) Suppose that X is a set and that fn : X ! R, n 2 {1, 2, 3, . . .}, and f : X ! R are maps. (a) Define what it means for the sequence (fn )n 1 to converge to f pointwise. Define what it means for the sequence (fn )n 1 to converge to f uniformly. (b) Prove or disprove: if (fn )n 1 converges to f uniformly then (fn )n 1 converges to f pointwise. (c) Prove or disprove: if (fn )n 1 converges to f pointwise then (fn )n 1 converges to f uniformly. Suppose now that X is a compact topological space, that each map fn : X ! R is continuous, and that fn+1 (x) fn (x) for all x 2 X and all n. Suppose further that (fn )n 1 converges to f pointwise and that f is continuous. Show that (fn )n 1 converges to f uniformly. (4) (a) Let X be a metric space. Define what it means for a map f : X ! X to be a contraction. (b) Let X be a complete metric space. Let k be a positive integer and let f : X ! X be a map such that the k-fold composition” f (k) = f | f ··· f {z } k times 1 2 MOCK FINAL EXAM QUESTIONS is a contraction. Show that f has a unique fixed point in X. Prove or disprove: or any x 2 X, the sequence f (n) (x) n 1 converges to the fixed point. (5) Let: B = f : [0, 1] ! R : f is bounded C = f : [0, 1] ! R : f is continuous C 1 = f : [0, 1] ! R : f is di↵erentiable and f 0 2 C . (a) Show that d(f, g) = sup |f (t) g(t)| t2[0,1] defines a metric on B. (b) Prove or disprove: C is closed in B. (c) Consider the maps: D : C1 ! C f 7 ! f0 Prove or disprove: (i) D is continuous; (ii) I is continuous. I: C ! C Z x f 7! f (t) dt 0 (6) (a) Let f : A ! B be a continuous map between topological spaces, let a0 2 A, and let b0 = f (a). (i) Define the fundamental group ⇡1 (A, a0 ). (ii) Define the map f? : ⇡1 (A, a0 ) ! ⇡1 (B, b0 ) and prove that it is well-defined. (b) Let X and Y be topological spaces, and let pX : X ⇥ Y ! X and pY : X ⇥ Y ! Y be the projection maps. We consider X ⇥Y as a topological space with the product topology. (i) What does it mean for a subset U ✓ X ⇥ Y to be open? (ii) Show that pX and pY are continuous. (iii) Show that, for a topological space Z, F : Z ! X ⇥ Y is continuous if and only if pX F and pY F are continuous. (iv) Hence, or otherwise, show that ⇡1 X ⇥ Y, (x0 , y0 ) ⇠ = ⇡1 (X, x0 ) ⇥ ⇡1 (Y, y0 ) as groups. (v) Show that (S 1 )n is homeomorphic to (S 1 )m if and only if n = m.