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Unformatted text preview: Math 185. Sample Answers to Problem Set #2 Page 32 6. First suppose that S is open. Using the books definition, this means that S contains none of its boundary points. So, if z S , then z is not a boundary point of S , and it is also not an exterior point of S (if z is an exterior point of S , then it has a neighborhood that is disjoint from S , but that neighborhood contains z itself, implying z / S ). Therefore z must be an interior point of S . Conversely, if each z S is an interior point, then it is not a boundary point, so S does not contain any of its boundary points. Thus S is open. 7. a . This is just the set { 1 , i } . It is a finite set, so it has no accumulation points (see problem 10). b . All points in this set are isolated, but they tend toward the origin, so { } is the set of accumulation points. c . This is the closed first quadrant: Re z 0 , Im z 0 . d . For even values of n , this is (1 + i ) n 1 n , with accumulation point 1 + i . For odd values of n this is (1 +...
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Lim
 Math

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