232sheet8 - ω ∈ F such that | z-α | ≥ | ω-α | for...

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B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 8 1 1. Let F be a closed set in C . Prove that if d ( ω,F ) = inf {| ω - z | : z F } = 0, then ω F . 2. Prove that a subset K of C is compact if and only if it is closed and bounded. 3. Cantor Intersection Theorem: Let F 1 be a nonempty, closed and bounded subset of C and let F 1 F 2 F 3 ⊇ ··· ⊇ F n ... be a sequence of nonempty closed sets. Then \ n N F n 6 = . 4. Nearest Point Theorem: Let F be a nonempty closed subset of C and α be a point in F c . Then there exists at least one point
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Unformatted text preview: ω ∈ F such that | z-α | ≥ | ω-α | , for all z ∈ F . 5. Prove that a line in C is a closed subset of C . 6. Is it true that if K is compact, then C \ K is connected? 7. Is { z ∈ C : | z | = 1 } compact? Is it connected? 8. Let K be a compact subset of C and ω ∈ C . Prove that the set K ω = { ω + z : z ∈ K } is also compact. 9. Prove that closed subsets of compact sets are compact. Notes 1 Please visit: http://www.math.boun.edu.tr/deptcourses/math232...
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This note was uploaded on 11/08/2011 for the course MATH 251 taught by Professor Gurel during the Spring '11 term at Boğaziçi University.

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