204Section52009

204Section52009 - Econ 204 Section 5 GSI: Hui Zheng Key...

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Econ 204 Section 5 GSI: Hui Zheng Key Words Open Set, Closed Set, Interior, Closure, Exterior, Boundary, Limits of Function, Continuity, Uniform Continuity, Lipschitz Functioins, Homemorphism Section 5.1 Open Set and Closed Set ( X;d ) be a metric space. A set A ± X is open if 8 x 2 A 9 " > 0 B " ( x ) ± A . A set C ± X is closed if X n C is open. int A : the interior of A , the largest open set contained in A (the union of all open sets contained in A ) A : the closure of A , the smallest closed set containing A (the intersection of all closed sets containing A ) ext A : the exterior of A , the largest open set contained in X n A @A : the boundary of A , ( X n A ) T A Lecture 4 Theorem 2: Let ( X;d ) collection of open sets is open. Lecture 4 Theorem 4: A set A in a metric space ( X;d ) is closed if and only if f x n g ² A , f x n g ! x 2 X ) x 2 A Openness and closedness depend on the underlying metric space as well as on the set. Theorem 2 is useful for proving that a set is open. Theorem 4 is useful for proving that a set is not closed. Unfortunately, using this theorem to prove that a set is closed is considerably more di¢ cult, but still possible in many cases. Example 5.1.1 Prove the statement or give a counterexample. Solution: False. Let A n = ( ³ 1 ; 1 n ) . \ 1 n =1 A n = ( ³ 1 ; 0] which is neither open nor closed. Notice that we can express a closed interval in R as the intersection of open intervals. [ a;b ] = \ 1 n =1 ( a ³ 1 n ; b + 1 n ) Example 5.1.2 Prove that the intersection of an arbitrary collection of closed and that the Solution: From de Morgan±s law we know that : ( A V B ) = : A W : B . Then the result Example 5.1.3 State whether the following sets are open, closed, both, or neither: 1. f ( x; 0) j x 2 (0 ; 1) g in R 2 2. f ( x;y;z ) j 0 ´ x + y ´ 1 ;z = 0 g in R 3 3. f 1 n j n 2 N g in R 4. f 1 n j n 2 N g in (0 ; 1 ) 5. Q in R Solution: 1. Neither 2. Closed 3. Neither 4. Closed 5. Neither Example 5.1.4 What is the closure of Q (the set of all rational numbers) Solution: Q = R . 1
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If 8 a;b 2 R a < b ) Q \ [ a;b ] = [ a;b ] , then Q \ R = [ n 2 Z f Q \ [ n; n +1] = [ n 2 Z [ n:n +1] = R : So our job is to show Q \ [ a;b ] = [ a;b ] . Consider the open interval ( a;b ) Q is closed, hence Q c is open. So ( a;b ) \ Q c is open. If ( a;b ) \ Q c 6 = ; , then there exists x 2 ( a;b ) \ Q c and " > 0 , such that ( x "; x + " ) ± ( a;b ) \ Q c . By the density of the rationals in R , there exists q 2 Q
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204Section52009 - Econ 204 Section 5 GSI: Hui Zheng Key...

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