22M:132Fall 07J. SimonSample Problem s for Exam IIProblem 1.a . S how that Rωwith the product topology is metrizable.b . Sho w that Rωwith the box topology is not metrizable.Problem 2.Let X be a metric space, with metric d, and let A be a no nempty subset of X.Define a function dA: X → R bydA(x) = inf{d(x, a) : a ∈ A}a. Prove the function dAis con tinuous.b. Suppose A and B are disjoint closed sets in the metric space X and assumein addition that A is compact. Prove there e xists ∆ > 0 such that f o r alla ∈ A, b ∈ B, d(a, b) ≥ ∆.c. Give an example to show we cannot omit the ass umption “A is compact” inpart (b); that i s, give an exam ple of a metric space X with disjoint closedsets A, B such that ∄ a distance ∆ > 0 between the sets.Problem 3.Let (fn) be a sequence of functions, fn: X → Y , where X is a topologica l space andY is a metric space; and let f : X → Y be some function.a. Define the statement “(fn) converges uniformly to f”.b. Prove: If (fn) converges uniformly to f and each fnis con tinuous, then f iscontinuous.c. Prove: If X is compact, and (fn) converges uniformly to f and each fniscontinuous and surjective, then f is surjective.Problem 4.a. Define quotient mapb. Explain why the followi ng function is not a quotient map (here S1is the unitcircle in R2):f : [0, 2π) → S1given by f(t) = (cos t, sin t)cJ. Simon, all rights reserved page 1Problem 5.Suppose f : X → Y is a surjec tive continuous function, X is compact, and Y isHausdorff. Prove that f is a quotient map.Problem 6.Suppose f : X → Y is continuous, 1-1, and surjective, X is compact, and Y isHausdorff. Prove f is a homeomorphism.Problem 7.Let X = S1, and define an equivalence relation o n X by saying ea ch point isequivalent to its antipode, i.e.(x, y) ∼ (−x, −y) .So each equivale nce class consists of exactly two points. Let Y be the quotient spaceX/ ∼.Prove Y is homeomorphic to S1.Hint: Define a func tion f : Y → S1that takes each eq uiva l ence class to a singlepoint of S1. Then use p revious problem(s?) to sho w f is a homeomorphis m.Problem 8.Show that the following properties of a space X are equivalent to each other:a. X = A ∪ B, where A and B are d i sjoint open se ts.b. X = A ∪ B, w here A and B are d i s joint closed sets.c. X = A ∪ B, wh ere neither of A, B intersects the closure of the other.Problem 9.Suppose X = A1∪ A2∪ . . ., where each set Aiis con nected and ∀i Ai∩ Ai+16= ∅.Prove X is connected.Problem 10.Suppose X is connected and f : X → Y is a surjective continuous map. Prove Y isconnected.cJ. Simon, all rights reserved page 2Problem 11.a. Define path connectedb. Prove the continuous image of a path connected space is pa th connected.c. Prove: If X and Y are pa th connected, then X × Y is path connected. (Statecarefully whatever lemma[s] you use.)Problem 12.Prove: If A is a conn ected subset of a s pace X, then the closure¯A is connected.Problem 13.(For this problem, assume we have proven that R is connected, and that finiteproducts of connected spa ces are co nnected.) The probl em has two related parts.Let A = {(an) ∈ Rω: an= 0 for all but fini tely many n}. Here Rωhas the producttopology.i) Show A is connected. (Hint: Write A as a nested union of connected sets.)ii) Use your result in part (i) to show that Rωis connec ted.Problem 14.Prove the interval [0, 1] is co nnected.Problem 15.a. Use Problems 9 and 14 to sho w that R1[with the standard topology] isconnected.b. Show that R1ℓis totally disconnected.Problem 16.Suppose f : S1→ R is a continuous function.Prove there e xists x ∈ S1such that f(x) = f (−x).Problem 17.Recall that a space X is locally path- connected if fo r each x ∈ X and eachneighborhood U of x, there exists a neighborhood V of x such that V ⊆ U and V ispath-connected.Prove: If X is connected and locally path-connected, then X is path-connected.(More generally, if U is a connected open subse t of a locally path-connected space,then U is path-connected.)cJ. Simon, all rights reserved page 3Problem 18.a. Define [connected] componentb. Prove that the components of any space are closed.c. Given an example to show that components do not have to be open.Problem 19.a. Define locally connectedb. Prove that X is locally connected if an only if for each open set U ⊆ X, eachcompon ent of U is o pen,Problem 20.a. Prove: If A is a closed subset of a compact space X then A i s compact.b. Prove: If A is a compact subset of a Hausdorff space X, then A is closed inX.Problem 21.a. If X is Hausdorff, x ∈ X, and A ⊆ X is a compact set that does not containx, then there exist disjoint neighborhoods U of x and V of A.b. If A, B are disjoint compact sets in a Hausdorff space X, then there existdisjoint neighbo rh oods U o f A and V of B.Problem 22.Prove: If X and Y are compact spaces , then the product X × Y is compact.Problem 23.a. Give an exampl e of an infinite collection of clo s ed sets in R1that has emptyintersection but such that each finite subcollection has nonempty intersection.b. Prove that in a compact space X, any collection of c losed sets with the finiteintersection property has nonempty intersec tion .Problem 24.Prove the interval [0, 1] ⊂ R1is compact.Problem 25.Prove that a set C ⊆ Rnis compact if an only if C is closed and bounded.cJ. Simon, all rights reserved page 4Problem 26.Suppose a compact metric space X is expre ssed as the union of two open sets,X = U ∪ V .Prove there e xists a number λ > 0 such that each subset of X hav i ng diameter < λis con tain ed in U or in V .Remark: At the risk of pointing out the obvious, this is not claiming that all the small setsare in U or all the small sets are in V . Some small sets end u p in U , others in V , andsome in both. Saying that a set is contained in U does not prevent i t from intersecting V ,maybe eve n being contained i n V as well as in U . If you think of a cartoon in which asmall set starts inside U and moves gradually to escape from U , by the time a part of thesmall set gets outside U , the set is entirely contained in V .Problem 27.a. Prove: If X is compact, then X is limit-point compact.b. Give an example of a space that is limit-point compact but not compact.Problem 28.a. If X is Hausdorff, x ∈ X, U a neighborhood …