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Unformatted text preview: Math 508 Exam 1 Jerry L. Kazdan October 12, 2006 12:00 – 1:20 Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5 points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems (60 points, 15 points each). Closed book, no calculators – but you may use one 3 00 × 5 00 card with notes. Part A: Examples (4 problems, 5 points each). Give an example of an infinite set in a metric space (perhaps R ) with the specified property. A–1. Bounded with exactly two limit points. Solution: The set { ( 1) n (1 + 1 n ) , n = 1 , 2 , 3 , . . . } in R . A–2. Containing all of its limit points. Solution: Lots of exmples: 1). The empty set. 2). All of R . 3). The point { } ∈ R . 4). The closed interval { ≤ x ≤ 1 in R } . A–3. Distinct points { x j , j = 1 , 2 , 3 , . . . } with x i 6 = x j for i 6 = j that is compact. Solution: The following subset of the real numbers: { } ∪ { 1 n , n = 1 , 2 , 3 , . . . } . A–4. Closed and bounded but not compact. Solution: The closed unit ball k x k ≤ 1 in ‘ 2 . The standard basis vectors e 1 = (1 , , , . . . ), e 2 = (0 , 1 , , , . . . ), etc have no convergent subsequence....
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 Fall '10
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 Topology, Sets, Metric space, CN, compact sets, A–1. Bounded

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