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Unformatted text preview: Section 8 Econ 204, GSI: Hui Zheng Key Words Separated, Connected, Correspondence, Upper Hemicontinuous Continuous, Lower Hemicontinuous Continuous, Closed-Valued, Compact-Valued, Closed Graph Section 8.1 Separated and Connected & Lecture 4 De&nition 4: A : the closure of A , the smallest closed set containing A (the intersec- tion of all closed sets containing A ) & Lecture 7 De&nition 1: Two sets A; B in a metric space are separated if A \ B = A \ B = ; . & Lecture 7 De&nition 1: A set in a metric space is connected if it cannot be written as the union of two nonempty separated sets. & Lecture 7 Theorem 2: A set S of real numbers is connected if and only if it is an interval. & Lecture 7 Theorem 3: Let X be a metric space, f : X ! Y continuous. If C ¡ X is connected, then f ( C ) is connected. Example 8.1.1 Let f S i g ;i 2 I; be a collection of connected subsets of a space X . Suppose there exists an i 2 I such that for each i 2 I , the sets S i and S i have non-empty intersection. Show that [ i 2 I S i is connected. Solution: Assume it is not true. Let U; V be nonempty separated sets in X with U [ V = [ i 2 I S i ; U \ V = U \ V = ; . We can show that for every i , U \ S i = S i or U \ S i = ; . To see this, note that U \ V = ; ) ( U \ S i ) \ ( V \ S i ) ¡ U \ V = ; . Similarly, ( U \ S i ) \ ( V \ S i ) = ; . Since S i is connected for every i , we have S i \ U = ; or S i \ U = S i . Similarly, for every i , we also have V \ S i = S i or V \ S i = ; . Furthermore, since U; V 6 = ; , 9 m; n such that U \ S m = S n and V \ S n = S n . Since S m \ S i 6 = ; ) U \ S i 6 = ; ) U \ S i = S i and similarly we have V \ S i = S i . Hence we have U \ V 6 = ; . Contradiction....
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.
- Summer '08