Math 132 MidtermAugust 2004[20-40pts] Answer all of the following 6 questions. You do not need to prove your answer forthese first 6 questions. Note for the first 5 questions, you are working in Z, the set of integers,(under various topologies including the subspace topology) and NOT the set of real numbers.1.) Let X = Z, then the set of limit points of (0, 3) ∩ X in X, [(0, 3) ∩ X]0=2.) Let X = Z, then the closure of (0, 3) ∩ X in X, (0, 3) ∩ X =3.) Let X = Z, then the interior of (0, 3) ∩ X in X, [(0, 3) ∩ X]o=4.) Suppose X = Z has the indiscrete topology, then the set of limit points of (0, 3) ∩ X in X,[(0, 3) ∪ X]0=5.) Suppose X = Z has the indiscrete topology, then the interior of (0, 3)∩X in X, [(0, 3)∩X]o=6.) If d is the discrete metric on R, the set of real numbers, thenBd(0, 1) = Bd(0, 2) =1[60-80pts] Prove 4 of the following 6. You may do more than these 4 problems for possiblepartial credit as discussed in class.Your best three answers are:Your fourth best answer is:1. Let X = {(x, y) ∈ R2| x = 0 or y = 0} and let W = {(x, y) ∈ R2| y = 0} where X and Ware subspaces of R2. Define f : X → W by f (x, y) = (x, 0).Is f continuous? Is f an open map? Is f a closed map. In each case, prove your answer.2. The fixed po int set F of a function f : X → X is the set F = {x ∈ X: f(x) = x}. Showthat if X is Hausdorff and f is continuous then F is a closed subset of X.3. Let f : X → Y be a bijective continuous function. If X is compact and Y is Hausdorff,prove that f is a homeomorphism. Give a specific example to show that f need not be ahomeomorphism if X is not compact. Also give a specific example to show that f need notbe a homeomorphism if Y is not Hausdorff.4. Show that X is compact if and only if every collection C of closed sets in X having the finiteintersection property, the intersection ∩C∈CC of all the elements of C is nonempty.5. Consider the product, uniform, and box topologies on Rω.In which topologies are the following functions from the set of real numbers to Rωcontinuous(prove your answer):f(x) = (x, x2, x3, ...)g(x) = (x, x, x, ...)k(x) = (0, 0, 0, ...)6. Let R∞be the subset of Rωconsisting of all sequences that are eventually zero. What is theclosure of R∞in Rωin the uniform topology. What is the closure of R∞in Rωin the boxtopology. Prove your
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