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Introduction to Topology Fall 2019 homework07-11nov19 1 MM/GS4016 Introduction to Topology Fall 2019 YOU WRITE SOLUTIONS ON SEPARATE SHEETS; IF YOU WRITE ANSWERS OR SOLUTION ON THIS SHEET, THEY WILL NOT BE EVALUATED. WRITE NAME (HANGUL), DEPT./YEAR, STUDENT #, ON EACH SHEET YOU SUBMIT! Homework-07 (17 problems, 340 points, 100%=80 points, until 150 points given) due Nov 11 Problem 1. (10 points) For the point p = (1, 2) and the line S = {3x − 2y = 5} in R2 calculate dist(p,S) for (the metric coming from the) H¨older 1-norm k . k1 and for the supremum norm k . k∞. Metrizability Problem 2. (50 points) (1) (5 points) Let A ⊂ [0, 1] be a set all of whose limit points (in the usual topology) are right. That is, if x ∈ Alim, then |∃ xn ր x (meaning xn → x and xn <xn+1 <x) with xn ∈ A. Prove that A is countable. .Hint: for x ∈ A, consider sup([0,x) ∩ A), and use this to exhibit an injection A ֒→ Q (2) (5 points) Prove that there is no function f : [0, 1] → (0, ∞) with lim f(x) = 0 for all x0 ∈ [0, 1]. x→x− 0 (3) (5 points) Modify the proof in class to conclude that [0, 1] × [0, 1) is not metrizable (with the dictionary order topology). (4) (15 points) Modify the proof to show that Rℓ (the lower limit topology) is not metrizable. Hint: prove that for a metric d there must be a function f : R → (0, ∞) with d(x,y) ≥ f(y) for x<y and that lim f(x) = 0 for all x0 ∈ R. x→x+ 0 (5) (10 points) Modify the proof to show that F(R, R) (with the product topology) is not metrizable. ′ Hint: Consider for x ր x0 f : R → R f ′ ≡ 0 ⊂ f : R → R f ′ ≡ 0 . R \{x } R \ [x , x0) (6) (5 points) Prove that there is a function f : Q → (0, ∞) with lim f(x) = 0 for all x0 ∈ Q. x→x0 (Thus the class proof fails for F(Q, R)= Rω. In fact, as you must know from your last HW, Rω is metrizable!) (7) (5 points) Use the previous construction to show that (with the usual topology on R) C(R, R) = { f : R → R | f continuous } is a discrete (and, as you saw in HW5, closed) subset of F(R, R)box. 2 homework07-11nov19 Separability and Lindel¨of property Let X be a topological space. X is called separable if X contains a dense countable subset (p.190 in book). Problem 3. (10 points) (4.30.13 p.192) Prove that if (X, A) contains an uncountable family of (pairwise) disjoint non-empty open sets (i.e. ∃O⊂A\{∅}, |O| >ω, ∀O1 6= O2 ∈ O : O1 ∩ O2 = ∅), then X is not separable. Problem 4. (15 points) (1) (6 points) Prove that the long line is not separable. (2) (9 points) Prove that SΩ is not separable. Problem 5. (10 points) (1) (6 points) (This is a step in the proof of the Borel-Lebesgue Covering theorem, thus do not use that theorem!) Let X be a sequentially compact metric space. Prove that X is separable. 1 Hint: consider coverings with ε-balls for ε = . n (2) (4 points) Is the converse true: if X is separable, is it compact? How about if X is separable and bounded? Give an argument or counterexample. 2 Remark: The ordered square I0 shows that ‘(covering or sequence) compact =⇒ separable’ fails for general topological spaces. The last part of the previous exercise should definitely not be too hard, and prompts a better question. Look at R. Problem 6. (5 points) Prove that a countable union of compact metric spaces is separable. That is, assume ∞ X is a metric space and X = Xi, such that Xi (with the restricted metric) are compact. i=1 Then prove that X is separable.S The interesting thing is: is the coverse true? In fact, the answer is ”no”: unlike R, not every separable (metric) space is a countable union of compact spaces. One can prove it thus: prove that if X is a separable (metric) space, then every A ⊂ X is separable. If A were a countable union of compact spaces, it would be a countable union of closed subsets of X. But for X = R, it is known from measure theory that there are sets A which are not Lebesgue measurable, while a countable union of closed sets are (they are called Fσ sets). We cannot do measure theory here. But I can ask you to prove the other step, which is also useful for itself. (You will see a much bigger example in your HW 10.) Problem 7. (30 points) homework07-11nov19 3 (1) (20 points) Prove if X is a separable metric space and A ⊂ X, then A (with the restricted metric) is separable. Hint: I did this by modifying the proof of the Borel-Lebesgue Covering theorem in class. (2) (5 points) Prove if X is a separable top. space, and A ⊂ X is open, then A (with the relative topology) is separable. (3) (5 points) Is it true that if X is a separable top. space, then any A ⊂ X is separable? Hint: Pb 10. Problem 8. (10 points) Prove that a metric space is separable if and only if every open cover has a countable subcover. Hint: Again, look at the Borel-Lebesgue proof. A topological space in which every open cover has a countable subcover is called a Lindel¨of space (p.192 of book). Problem 9. (20 points) We define the countable-complement-open (CCO) topology on R given by open sets O \ U where O is open in the Euclidean topology, and U is countable. (1) (10 points) Prove that this is a topology. Hint: for the union property, argue and use that any open set in R is Lindel¨of. (2) (10 points) Prove that this topology is Hausdorff but not regular. Thus T2 =⇒ T3 is false in general. Also, this example shows that T3 (and T4) is not always preserved under refinement. Problem 10. (40 points) (1) (5 points) Prove that if (X, d) is a separable metric space, then |X|≤ℵ1. (2) (5 points) Give an example of a metric space X with |X|≤ℵ1 which is not separable. (3) (10 points) Prove that if (X, d) is a separable metric space, and A ⊂ X is discrete, then A is countable. 2 (4) (10 points) Let the Sorgenfrey Plane (p.191 in book) be L = Rℓ = Rℓ ×Rℓ (with product topology). Prove that L is separable, but the antidiagonal line {(x, −x) : x ∈ R} is discrete and uncountable. 2 Thus L is a counterexample for part 3 for a general top. space. You see also another proof that (Rℓ and hence) Rℓ is not metrizable. (5) (5 points) What is the cardinality of { A ⊂ R : A discrete } ? (R with the usual topology) (6) (5 points) Conclude that C(R, R) has the cardinality of the continuum |R| = ℵ1. Problem 11. (30 points) Prove the converse to Pb 3 in a metric space X. Theorem 1. Assume in X every family of (pairwise) disjoint non-empty open sets is countable. Then X is separable. 4 homework07-11nov19 (1) (6 points) Call Mε ⊂ X an ε-grid if ∀x 6= y ∈ Mε, d(x,y) ≥ ε. Prove that in X every ε-grid is countable. (2) (8 points) Prove that for every ε, there is a maximal ε-grid. (Hint: Zorn lemma.) (3) (7 points) Prove that every maximal ε-grid is an ε-net, and complete the proof of the theorem. (4) (9 points) Consider on R the CCO topology. Prove that for this space X the theorem is false. (We have seen that this topology is not regular, so it is surely not metrizable.) Problem 12. (15 points) Modify the solution to Pb 11 to show that ‘every discrete subset is countable =⇒ the space is separable’ is true for a metric space, but not for a general T2 space. (This treats the converse of Pb 10 parts 3,4.) Problem 13. (15 points) (1) (5 points) Is Rℵ1 = F(R, R) separable with the box topology? Argue why or why not. (2) (5 points) How about the uniform topology? (3) (5 points) The product topology? Hint: Lagrange interpolation (Remark: if you combine this with Pb 10, you have yet another proof that F(R, R)prod is not metrizable!) Problem 14. (15 points) (1) (9 points) Prove that Rℓ is Lindel¨of, in the following way. Prove for [0, 1). Let O be an open cover with no countable subcover, consider inf { b ∈ [0, 1] : no countable subfamily of O covers [0, b) } and derive a contradiction. 2 (2) (6 points) Is Rℓ Lindel¨of? Why or why not? Either give a proof, or write down an open cover for which you argue that it has no countable subcover. Problem 15. (25 points) (1) (12 points) Prove that SΩ is not a Lindel¨of space (i.e., it has an open cover with no countable subcover, 7 points), and conclude that neither is the long line (5 points). ∗ ′ (2) (13 points) Is SΩ Lindel¨of? (2 points) Now consider SΩ to be SΩ with the topology A coming from adding to the order topology basis B = { [a0, a) } ∪ { (a, b) } ∪ { (b, Ω] } ′ ′ ′ ∗ the set {Ω}, i.e., A = A(B ) for B = B ∪ {{Ω}}. Prove that SΩ and SΩ have the same covergence ′ behavior, i.e., xn → x in A = A(B) if and only if xn → x in A (7 points), but the one topology is (covering) compact and the other not (even) Lindel¨of.
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