Homework 4 Solutions Problem 13.4. Proposition. If {Tα } is a family of topologies on X, then T Tα is a topology on X. Proof. Since ∅ and X are in each Tα , they must also be elements of then C ⊂ Tα for all α; it follows that is a finite subcollection of T S Tα , then C ∈ Tα for all α, and hence T T Tα . Next, if C ⊂ S C∈ C ∈ Tα for all α, and therefore T T T Tα , Tα . Finally, if C C∈ T Tα . It is not true in general that the union of two topologies is a topology. For example, the union T1 ∪ T2 = {∅, X, {a}, {a, b}, {b, c}} of the two topologies from part (c) is not a topology, since {a, b}, {b, c} ∈ T1 ∪ T2 but {a, b} ∩ {b, c} = {b} ∈ / T1 ∪ T2 . Proposition. Let {Tα } be a family of topologies on X. Then there exists a unique smallest topology on X containing all the collections Tα , and a unique largest topology contained in all Tα . Proof. The unique largest topology contained in all the Tα is simply the intersection For the other statement, observe that the family of all topologies on X that contain T S Tα . Tα is nonempty, since it includes the discrete topology on X. Then the intersection of this family is the unique smallest topology that contains every Tα . For part (c), the smallest topology containing both T1 and T2 is {∅, X, {a}, {b}, {a, b}, {b, c}}, and the largest topology contained in T1 and T2 is the intersection T1 ∩ T2 = {∅, X, {a}}. 1 Problem 13.8. Proposition. The countable collection B = {(a, b) | a < b, a and b rational} is a basis for the standard topology on R. Proof. Let U ⊂ R be open, and let x ∈ U . Since the open intervals are a basis for the topology on R, there exists an open interval (a, b) such that x ∈ (a, b) ⊂ U . Then a < x < b, so there exist rational numbers q and r such that a < q < x < r < b. Then (q, r) ∈ B and x ∈ (q, r) ⊂ U , so B is a basis for the standard topology on R by Lemma 13.2. Proposition. The basis C = {[a, b) | a < b, a and b rational} generates a topology on R different from the lower limit topology. √ √ Proof. The set [ 2, 2) is open in the lower-limit topology. However, since 2 is irrational, there does not exist an [a, b) ∈ C for which √ √ 2 ∈ [a, b) ⊂ [ 2, 2). Problem 16.4. 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 . Proposition. If X and Y are topological spaces, then the projections π1 : X × Y → X and π2 : X × Y → Y are open maps. Proof. Let U be an open set in X × Y . Then U is a union of basic open sets in X × Y , say U = π1 (U ) = S S α Vα × Wα , where each Vα is open in X and each Wα is open on Y . Then α π1 (Vα × Wα ) = S α Vα , which is open in X, and similarly π2 (U ) is open in Y . 2 Problem 16.9. Proposition. The dictionary order topology on the set R × R is the same as the product topology Rd × R, where Rd denotes R in the discrete topology. Proof. Because singleton sets are a basis for the topology on Rd , the collection B = {{a} × (b, c) | a, b, c ∈ R and b < c} is a basis for the product topology on Rd × R. Each of these sets is open in the dictionary order topology, since {a}×(b, c) = (a×b, a×c), and thus the topology on Rd ×R is contained in the dictionary order topology. For the other direction, observe that every ray in the dictionary order topology is open in Rd × R. In particular, (a × b, ∞) = {a} × (b, ∞) ∪ (a, ∞) × R and (−∞, a × b) = (∞, a) × R ∪ {a} × (−∞, b) . Since the rays are a subbasis for the dictionary order topology, it follows that the dictionary order topology is contained in the product topology on Rd × R. The dictionary order topology on R × R contains the standard topology. In particular, every basic open rectangle (a, b) × (c, d) in the standard topology is also open in Rd × R, since (a, b) is open in Rd and (c, d) is open in R. This containment of topologies is strict, e.g. {0} × (0, 1) is open in the dictionary order topology but not in the standard topology. 3
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