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Topology Hints and Solutions to Munkres’s Book ∗ DongPhD (Editor) DongPhD Problem Books Series υo.1 Available at http://dongphd.blogspot.com ∗ James R. Munkres “Topology”-Prentice Hall, 2 ed. | 2000 | ISBN: 0139254951 1 DongPhD 2 Exercises “Say what you know, do what you must, come what may.”- Sofia Kovalevskaya Topological spaces, basis for a topology Ex 13.1. Let X be a topological space; let A be a subset of X. Suppose that for each x ∈ A there is an open set U containing x such that U ⊂ A. Show that A is open in X. Ex 13.2. Ex 13.3. Ex 13.4. (a) If {τ α } is a family of topologies on X, show that ∩{τ α } is a topology on X. Is ∪{τ α } a topology on X? (b) Let {τ α } be a family of topologies on X. Show that there is a unique smallest topology on X containing all the collections {τ α }, and a unique largest topology contained in all {τ α }. (c) If X = {a, b, c}, let τ 1 = {∅, X, {a}, {a, b}} and τ 2 = {∅, X, {a}, {c, b}}. Find the smallest topology containing τ 1 and τ 2 , and the largest topology contained in τ 1 and τ 2 . Ex 13.5. Show that if A is a basis for a topology on X, then the topology generated by A equals the intersection of all topologies on X that contain A. Prove the same if A is a subbasis. Ex 13.6. Show that the topologies of R l and R K - are not comparable. Ex 13.7. Consider the following topologies on R: τ 1 = the standard topology, τ 2 = the topology of R K - , τ 3 = the finite complement topology, τ 4 = the upper limit topology, having all sets (a, b] as basis, τ 5 = the topology having all sets (−∞, a) = {x|x < a} as basis. Determine, for each of these topologies, which of the others it contains. http://dongphd.blogspot.com DongPhD 3 Ex 13.8. (a) Show that the countable collection B = {(a, b) | a < b, a and b rational} is a basis that generates t he standard topology on R. (b) Show that the collection C = {[a,b)|a<b, a and b rational} is a basis that generates a topology different from the lower limit topology on R Subspace topology Ex 16.9. Show that if Y is a subspace of X, and A is a subset of Y , then the topology A inherits as a subspace of Y is the same as the topology it inherits as a subspace of X. Ex 16.10. Ex 16.11. Consider the set Y = [−1, 1] as a subspace of R. Which of the following sets are open in Y ? Which are open in R? A = {x| 1 2 < |x| < 1} B = {x| 1 2 < |x| ≤< 1} C = {x| 1 2 ≤ |x| < 1} D = {x| 1 2 ≤ |x| ≤ 1} E = {x|0 < |x| < 1 and 1 x /∈ Z + } Ex 16.12. A map f : X → Y is said to be an open map if for every open set U of X, the set f(U) is open in Y . Show that π 1 : X ×Y → X and π 2 : X × Y → Y are open maps. Ex 16.13. http://dongphd.blogspot.com DongPhD 4 Ex 16.14. Show that the countable collection {(a, b) × (c, d)|a < b and c < d and a, b, c, d arerational} Ex 16.15. Let X be an ordered set. If Y is a proper subset of Y that is convex in X, does it follow that Y is an interval or a ray in X? Ex 16.16. Ex 16.17. Show that the dictionary order topology on the set R × R is the same as the product topology R d × R, where R d denotes R in the discrete topology. Compare this topology with the standard topology on R 2 . Closed Sets and Limit Points Ex 17.18. Ex 17.19. Ex 17.20. Show that if A is closed in X and B is closed in Y , then A × B is closed in X × Y . Ex 17.21. Ex 17.22. Ex 17.23. Let A, B and A α denote subsets of a space X. Prove the following: (a) If A ⊂ B then A ⊂ B (b) A ∪ B = A ∪ B (c) ∪A α ⊃ ∪A α , give an example where equality fails. Ex 17.24. Ex 17.25. Let A, B and A α denote subsets of a space X. Determine whether the following equations hold; if an equality fails, determine whether one of the inclusions ⊃ or ⊂ holds. http://dongphd.blogspot.com DongPhD 5 (a) A ∩ B = A ∩ B (b) ∩A α ⊃ ∩A α , (c) A \ B = A \ B Ex 17.26. Let A ⊂ X andB ⊂ Y . Show that in the space X × Y , A × B = A × B Ex 17.27. Show that every order topology is Hausdorff. Ex 17.28. Show that the product of two Hausdorff spaces is Hausdorff. Ex 17.29. Show that a subspace of a Hausdorff space is Hausdorff. Ex 17.30. Show that X is Hausdorff if and only if the diagonal A = {(x, x)|x ∈ X} is closed in X × X. Ex 17.31. In the finite complement topology on R, to what point or points does the sequence x n = 1 n converge? Ex 17.32. (Kuratowski) Consider the collection of all subsets A of the topological space X. The operations of closure A → A and complemen- tation A → X \ A are func- tions from this collection to itself. (a) Show that starting with a given set A, one can form no more than 14 distinct sets by applying these two operations successively. (b) Find a subset A of R (in its usual topology) for which the maxi- mum of 14 is obtained. Continuous Functions Ex 17.33. Prove that for functions f : R → R, the  − δ definition of continuity implies the open set definition. Ex 17.34. Suppose that f : X → Y is continuous. If x is a limit point of the subset A of X, is it necessarily true that f(x) is a limit point of f(A)? Ex 17.35. http://dongphd.blogspot.com DongPhD 6 Ex 17.36. Ex 17.37. Ex 17.38. Find a function /f : R → R that is continuous at precisely one point. Ex 17.39. (a) Suppose that f : R → R is "continuous from the right," that is, lim x→a + f(x) = f(a) Show that f is continuous when considered as a function from R l to R. (b) Can you conjecture what functions f : R → R are continuous when considered as maps from R to R l ? As maps from R l to R l ? Ex 17.40. Let Y be an ordered set in the order topology. Let f, g : X → Y be continuous. (a) Show that the set {x|f(x) < g(x)} is closed in X. (b) Let h : X → Y be the function h(x) = min{f(x), g(x)}. Show that h is continuous. Ex 17.41. Ex 17.42. Let f : A → B and g : C → D be continuous functions. Let us define a map f × g : A × C → B × D by the equation (f × g)(a × c) = f(a) × g(c). Show that f × g is continuous. Ex 17.43. Ex 17.44. Ex 17.45. Let A ⊂ X; let f : A → Y be continuous; let Y be Hausdorff. Show that if f may be extended to a continuous function g : A → Y , then g is uniquely determined by f. http://dongphd.blogspot.com DongPhD 7 The Quotient Topology Ex 22.46. Ex 22.47. (a) Let p : X → Y be a continuous map. Show that if there is a continuous map f : Y → Xsuch that p ◦ f equals the identity map of Y , then p is a quotient map. (b) If A ⊂ X, a retraction of X onto A is a continuous map r : X → A such that r(a) = a for each a ∈ A. Show that a retraction is a quotient map. Ex 22.48. Let π 1 : R × R → R be projection on the first coordinate. Let A be the subspace of R × R consisting of all points x × y for which either x > 0 or y = 0 (or both); let q : A → Rbe obtained by restricting π 1 . Show that q is a quotient map that is neither open nor closed. Ex 22.49. Ex 22.50. Let p : X × Y be an open map. Show that if A is open in X, then the map q : A → p(A) obtained by restricting p is an open map. Compact Spaces Ex 26.51. (a) Let τ and τ  be two topologies on the set X; suppose that τ  ⊃ τ. What does compactness of X under one of these topologies imply about compactness under the other? (b) Show that if X is compact Hausdorff under both τ and τ  , then either τ and τ  are equal or they are not comparable. Ex 26.52. (a) Show that in the finite complement topology on R, every subspace is compact. (b) If R has the topology consisting of all sets A such that R \ A is either countable or all of R, is [0, 1] a compact subspace? Ex 26.53. Show that a finite union of compact subspaces of X is com- pact. Ex 26.54. http://dongphd.blogspot.com DongPhD 8 Ex 26.55. Let A and B be disjoint compact subspaces of the Hausdorff space X. Show that there exist disjoint open sets U and V containing A and B, respectively. Ex 26.56. Show that if f : X → Y is continuous, where X is compact and Y is Hausdorff, then f is a closed map (that is, f carries closed sets to closed sets). Ex 26.57. Show that if Y is compact, then the projection π 1 : X ×Y → X is a closed map. Ex 26.58. Let f : X → Y ; let Y be compact Hausdorff. Then f is continuous if and only if the graph of f is closed in X × Y . Ex 26.59. Ex 26.60. Ex 26.61. Ex 26.62. Let p : X → Y be a closed continuous surjective map such that p −1 ({y}) is compact, for each y ∈ Y . (Such a map is called aperfect map.) Show that if Y is compact, then X is compact. Compact Subspaces of the Real Line Ex 26.63. Prove that if X is an ordered set in which every closed interval is compact, then X has the least upper bound property. Ex 26.64. Ex 26.65. Recall that R K denotes R in the K -topology. (a) Show that [0, 1] is not compact as a subspace of R K (b) Show that R K is connected. (c) Show that R K is not path connected. Ex 26.66. Ex 26.67. Let X be a compact Hausdorff space; let A n be a count- able collection of closed sets of X. Show that if each set A, has empty interior in X, then the union ∪A n has empty interior in X. http://dongphd.blogspot.com DongPhD 9 Local Compactness Ex 29.68. Show that the rationale Q are not locally compact. Ex 29.69. Let {X α } be an indexed family of nonempty spaces. (a) Show that if  X α is locally compact, then each X α is locally compact and X α is compact for all but finitely many values of or. (b) Prove the converse, assuming the Tychonoff theorem. Ex 29.70. Let X be a locally compact space. If f : X → Y is con- tinuous, does it follow that f(X) is locally compact? What if f is both continuous and open? Justify your answer. The Countability Axioms Ex 30.71. Ex 30.72. Ex 30.73. Let X have a countable basis; let A be an uncountable subset of X. Show that uncountably many points of A are limit points of A. Ex 30.74. Show that every compact metrizable space X has a countable basis. Ex 30.75. (a) Show that every metrizable space with a countable dense subset has a countable basis, (b) Show that every metrizable Lindelof space has a countable basis. Ex 30.76. Show that R l and I 2 0 are not metrizable. Ex 30.77. Which of our four countability axioms does S ω satisfy? What about S ω ? Ex 30.78. Ex 30.79. Let A be a closed subspace of X. Show that if X is Lindelof, then A is Lindelof. Show by example that if X has a countable dense subset, A need not have a countable dense subset. http://dongphd.blogspot.com DongPhD 10 Ex 30.80. Ex 30.81. Ex 30.82. Let f : X → Y be a continuous open map. Show that if X satisfies the first or the second countability axiom, then f(X) satisfies the same axiom. Ex 30.83. Show that if X has a countable dense subset, every collec- tion of disjoint open sets in X is countable. The Separation Axioms Ex 31.84. Show that if X is regular, every pair of points of X have neighborhoods whose closures are disjoint. Ex 31.85. Show that if X is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint. Ex 31.86. Show that every order topology is regular. Ex 31.87. Ex 31.88. Let f, g : X → Y be continuous; assume that Y is Haus- dorff. Show that {x|f(x) = g(x)} is closed in X. Ex 31.89. Let p : X → Y be a closed continuous surjective map. Show that if X is normal, then so is Y . Ex 31.90. Let p : X → Y be a closed continuous surjective map such that p −1 ({y}) is compact, for each y ∈ Y . (Such a map is called aperfect map.) (a) Show that if X is Hausdorff, then so is Y . (b) Show that if X is regular, then so is Y . (c) Show that if X is locally compact, then so is Y . (d) Show that if X is second-countable, then so is Y . http://dongphd.blogspot.com [...]... result Finally note that the set of topologies on the set X is partially ordered, c.f ex 11.2, under inclusion From the lemma we conclude that the compact Hausdorff topologies on X are minimal elements in the set of all Hausdorff topologies on X Ex 26.2 (Morten Poulsen) (a) The result follows from the following lemma Lemma 2 If the set X is equipped with the finite complement topology then every subspace of... that Tc is a topology on X This topology is called the countable complement topology Lemma 3 The compact subspaces of X are exactly the finite subspaces Proof Suppose A is infinite Let B = {b1 , b2 , } be a countable subset of A Set An = (X − B) ∪ {b1 , , bn } Note that {An } is an open covering of A with no finite subcovering The lemma shows that [0, 1] ⊂ R in the countable complement topology is... (B) is a basis for a topology Tf (B) on f (X) This topology is coarser than the topology on f (X) since the basis elements are open in f (X) Conversely, let f (x) ∈ V ∩ f (X) where V is open in Y Choose a basis element B such that x ∈ B ⊂ f −1 (V ) Then f (x) ∈ f (B) ⊂ V ∩ f (X) This shows that all open subsets of f (X) are in Tf (B) We conclude that f (B) is a basis for the topology on f (X) We... contained in any compact subspace of RK [Thm 28.1] (b) The subspaces (−∞, 0) and (0, +∞) inherit their standard topologies, so they are connected Then also their closures, (−∞, 0] and [0, +∞) and their union, RK , are also connected [Thm 23.4, Thm 23.3] (c) Since the topology RK is finer than the standard topology [Lemma 13.4] on R we have U is connected in RK Ex 23.1 ⇒ U is connected in R Thm 24.1 ⇔ U is convex... Munkres §26 Ex 26.1 (Morten Poulsen) (a) Let T and T be two topologies on the set X Suppose T ⊃ T If (X, T ) is compact then (X, T ) is compact: Clear, since every open covering if (X, T ) is an open covering in (X, T ) If (X, T ) is compact then (X, T ) is in general not compact: Consider [0, 1] in the standard topology and the discrete topology (b) Lemma 1 If (X, T ) and (X, T ) are compact Hausdorff... is a quotient map [Example 7, p 143] but is is true that a product of two open maps is an open map.) This shows [Thm 22.2] that ϕ/H is continuous and hence [SupplEx 22.1] G/H is a topological group SupplEx 22.7 Let G be a topological group (a) Any neighborhood of U of e contains a symmetric neighborhood V ⊂ U such that V V ⊂ U By continuity of (x, y) → xy, there is a neighborhood W1 of e such that... space is normal Ex 32.92 Ex 32.93 Show that every locally compact Hausdorff space is regular Ex 32.94 Show that every regular Lindelof space is normal Ex 32.95 Is Rω normal in the product topology? In the uniform topology? Ex 32.96 A space X is said to be completely normal if every subspace of X is normal Show that X is completely normal if and only if for every pair A, B of separated sets in X (that... nontrivially, the point (x, y) lies in the closure of A × B [Thm 17.5] This shows that A × B ⊂ A × B 1 2 Ex 17.10 (Morten Poulsen) Theorem 1 Every order topology is Hausdorff Proof Let (X, ≤) be a simply ordered set Let X be equipped with the order topology induced by the simple order Furthermore let a and b be two distinct points in X, may assume that a < b Let A = { x ∈ X | a < x < b }, i.e the set... V , U and V open in X, for the product topology, such that a × b ∈ U × V ⊂ ∆c Since U × V ⊂ ∆c it follows that U ∩ V = ∅ Hence U and V are open sets such that a ∈ U , b ∈ V and U ∩ V = ∅, i.e X is Hausdorff Ex 17.14 (Morten Poulsen) The sequence converges to every real number, by the following result Theorem 5 Let X be a set equipped with the finite complement topology If (xn )n∈Z+ is an infinite sequence... 22.3 Let G be a topological group and H a subgroup Let ϕG : G × G → G be the map ϕG (x, y) = xy −1 and ϕH : H × H → H the corresponding map for the subgroup H Since these maps are related by the commutative diagram ϕG G×G O ? H ×H ϕH /G O ? /H and ϕG is continuous, also ϕH is continuous [Thm 18.2] Moreover, any subspace of a T1 -space is a T1 -space, so H is a T1 -space Thus H is a topological group . following topologies on R: τ 1 = the standard topology, τ 2 = the topology of R K - , τ 3 = the finite complement topology, τ 4 = the upper limit topology, having all sets (a, b] as basis, τ 5 = the topology. a family of topologies on X, show that ∩{τ α } is a topology on X. Is ∪{τ α } a topology on X? (b) Let {τ α } be a family of topologies on X. Show that there is a unique smallest topology on. dictionary order topology on the set R × R is the same as the product topology R d × R, where R d denotes R in the discrete topology. Compare this topology with the standard topology on R 2 . Closed