22M:132Fall 07J. SimonInformation and Sample ProblemsforExam IInstructions. This is a ”closed book” exam; you should have no books, scratchpaper, or other written material ava ilable during the exam. Do all your work in theexam b ooklet provided.***There are 8 problems; each is worth 10 points, so the total is 80 points.Advice. The exam is written under the assumption that you know the materialand can work the problems efficiently. So if you get stuck on something, don’t spenda lot of time o n it until after you’ve worked the problems yo u can do more quickly.NOTE NOTE NOTE. The first 8 pro blems a re a n actual exam I gave previouslyin 22M:132. For our exam, I do not expect to include the t opic of ”well-ordering”(so Problem 2 below is moot). But students asked to see an ”actual” exam - so hereit is.The rest of the problems... are samples. I may send out more sample problemslater (e.g. Friday or Saturday). My standard commitment is that “most of theproblems on the exam will be taken from the list of sample pro blems slightmodifications of the samples”. I reserve the right to have problems on the examthat are not represented in the samples.Problem 1. It took some work to prove that Z, Z × Z, and Q are countable. Usingthese facts, and wh atever general theorems you need about countable sets, explainhow we know that eac h of the following sets A is countable.• a) A = {(p, q) ∈ Q × Q such that p < q}• b) A = Z × {1 , 2} .• c) Let rZ denote the set of all multiples r · z, z ∈ Z. DefineA = π Z ∪ π2Z ∪ π3Z ∪ π4Z ∪ . . .Problem 2. Suppose X is an uncountable set with a well- ordering “<”. Show thereexists a subset Y ⊆ X with the following pro perty:∀y ∈ Y, {s ∈ Y | s < y} is countable.cJ. Simon, all rights reserved page 1Problem 3. Suppose (X, T ) and (Y, T′) are topological spaces. LetπX: X × Y → X and πY: X × Y → Y be the coordinate projection functions, that isπX(x, y) = x and πY(x, y) = y.Prove that the product topology is the coarsest topology on X × Y that makes bothprojections continuious.Problem 4. Let X = Rℓ× Rℓwith the product topology. Let A = (0, 1) × (0, 1) ⊆ X.(a) F i nd int(A).(b) Find the set of limit points of A.(c) FindA.(d) F i nd bd(A).Problem 5. Let (X, T ) be a topological space, with A ⊆ X. Prove:bd(A) = ∅ ⇐⇒ A is both open and closed.(Make it clear what definition or theorem(s ) you are using. )Problem 6. Let (X, T ) be a topological space, with Y ⊆ X; give Y the subspacetopology TY. Here are two false“theorems”. Give counterexamples to the“theorems” as stated, and change the statements to make them valid theorem s .(a) I f U ⊆ Y is open in Y (i.e. U ∈ TY), then U is open in X.(b) I f C ⊆ Y is closed in Y then C is closed in X.Problem 7. Suppose (X, <) and (X′, <′) are simply ordered (i . e. linearly ordered)sets. Give each set the order topology. Suppose f : X → X′is surjective andmonotonic, that is∀x, y ∈ X, x < y =⇒ f(x) <′f(y).Show that f i s a homeomorphism.cJ. Simon, all rights reserved page 2Problem 8. We are going to define an unusual topology on the set of naturalnumbers, Z+, by specifying a basis. For each j ∈ Z+, letUj= { n ∈ Z+| n is divi sible by j} .For example, U3= {3, 6, 9, 12, . . . }.• (a) Show the set of these Un’s is a basis for a topology on Z+. (Hint: firstobserve that r ∈ Up=⇒ Ur⊆ Up.)• (b) Show the topology is not Hausdorff.END OF EXAM FROM A PRIOR YEARADDITIONAL SAMPLE PROBLEMSProblem 9. Definitions: You should know all the de finitions we have discussedin class (more precisely, all the definitions in all the sections of the text we havecovered). The specific topologies you should be able to recogn i ze/work with are:• The usual (i.e. standa rd) topology on Rn• The discrete and indiscrete topologies• The ord er topology on an ordered set• The product topo l ogy (finite or infinite products)• The box topology on an in finite product• The lower-limit topology on R. When we want to use this, we write Rℓ.• The counter-finite (i.e. finite-complemen ts) topology on R.Problem 10. (list of “HW” prob l ems from Section 19)• Page 118 Ex 2• Page 118 Ex 6• Page 118 Ex 7• Page 118 Ex 10a (read b), cProblem 11. Assuming Theorems 7.1, 7.2, and 7.3, be able to prove various setsare countable. This includes specifi c sets such as any part(s) of Page 51 Ex 5, andalso ge neral results such as• Theo rem 7.6• Theo rem 7.7cJ. Simon, all rights reserved page 3Problem 12. (this proposition, and the next, are how I think we should navigatethrough Section 13)If B is a collection of subsets of X satisfying the two properties in the Definitionon page 7 8, then the set of all unions of elements of B is a topology for X.Problem 13. (and for subbases...)If S is a collection of subsets of X whose union is all of X, then the set of all unionsof finite intersections of elements of S is a topology for X.Problem 14. • Page 83 Ex 1• Page 83 Ex 4 a,b• Page 83 Ex 7• (variation on Page 83 Ex 6) Decide whether or not the lower-limit topologyon R a nd the finite-complements topology on R are comparable. Explain youranswer.Problem 15. Suppo se (X, T )is a topological space and S is a countable subbasis forT . Decide whether or not there must exi s t a countable basis for T ; prove or givecounter-example.Problem 16. If X × Y has the product topol ogy, prove• (a) The projection πX: X × Y → X is continuous.• (b) The projection πX: X × Y → X is an open map.Problem 17. State a nd prove (or giv e counterexample[s]) theorems analogous tothe previo us ones, but now for infinite products wi th the poduct topology; for infiniteproducts with the box topology.Problem 18. • Page 92 Ex 8• Page 92 Ex 9Problem 19. Using the text definition of closure, prove that a set A ⊆ X is closed⇐⇒ A contains all of its limit points. (This is an immediate coro llar y of Theorem17.6, so don’t just quote that theorem; view this problem as an alternate way tostate Theorem 1 7.6 and prove it “from scratch”.)Problem 20. Page 95 Theorem 17.4Problem 21. Page 101 Exercises 6, 8, 9cJ. Simon, all rights reserved page 4Problem 22. Prove that X × Y is Hausdorff ⇐⇒ each of X and Y is Hausdorff.Problem 23. Decide if the theorem i n the previous theorem still works for infiniteproducts in the product topology? in the box topol ogy?? (so there are 4 situations todecide: i mplication “if” or implication