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232sheet7

# 232sheet7 - B U Department of Mathematics Math 232...

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B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 7 1 In what follows, ”open” and ”closed” means ”open with respect to C ” and ”closed with respect to C ”, respectively; D ( z, r ) stands for the open disc. 1. Show that arbitrary intersection of open sets is not necessarily open. 2. Give an example of a subset of C which is neither open, nor closed. 3. Show that the countable union of closed sets is not necessarily closed. 4. Let S = D T where D = D (0 , 1) and T = { z C | Re z, Im z Q } . Show that every point in S is a limit point. 5. Is there a subset S of C such that S has empty interior and ¯ S = C ? 6. Prove that a finite subset of C does not have any limit points. 7. Prove that a) a set is open if and only if it contains none of its boundary points. b) a set is closed if and only if it contains all of its boundary points. 8. Let S = { z C | 0 < Re z 1 } . Is S open?

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232sheet7 - B U Department of Mathematics Math 232...

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