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508F08Ex1s(1) - Exam 1 Math 508 Jerry L Kazdan 10:30 11:50...

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Math 508 Exam 1 Jerry L. Kazdan October 16, 2008 10:30 – 11:50 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 or computers– but you may use one 3 00 × 5 00 card with notes on both sides. Part A: Examples (4 problems, 5 points each). Give an example having the specified property. A–1. A metric space that contains all but one of its limit points. A–2. An open cover of { x R : 0 < x 1 } that has no finite sub-cover. A–3. A metric space having a bounded infinite sequence with no convergent subsequence. A–4. A metric space that is not complete. Part B: Classify sets (20 points) For each of the following sets, circle the listed properties it has: a) { 1 - 1 n R , n = 1 , 2 , 3 , . . . } open closed bounded compact countable b) { 1 } ∪ { 1 + ( - 1) n n R , n = 1 , 2 , 3 , . . . } open closed bounded compact countable c) { ( x, y ) R 2 : 0 y - x 1 } open closed bounded compact countable d) { ( x, y ) R 2 : 0 < x 2 + y 2 } open closed bounded compact countable e) { ( x, y ) R 2 : x > 1 , y < 1 x }
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