204Lecture42006

204Lecture42006 - Economics 204 Lecture 4August 3, 2006...

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Unformatted text preview: Economics 204 Lecture 4August 3, 2006 Revised 8/4/06, Revisions indicated by ** Section 2.4, Open and Closed Sets Definition 1 Let ( X, d ) be a metric space. A set A X is open if x A > B ( x ) A A set C X is closed if X \ C is open. Example: ( a, b ) is open in the metric space E 1 ( R with the usual Euclidean metric). Given x ( a, b ), a < x < b . Let = min { x a, b x } > Then y B ( x ) y ( x , x + ) ( x ( x a ) , x + ( b x )) = ( a, b ) so B ( x ) ( a, b ), so ( a, b ) is open. Notice that depends on x ; in particular, gets smaller as x nears the boundary of the set. Example: In E 1 , [ a, b ] is closed. R \ [ a, b ] = ( , a ) ( b, )** is a union of two open sets, which must be open . . . . Example: In the metric space [0 , 1], [0 , 1] is open. With [0 , 1] as the underlying metric space, B (0) = { x [0 , 1] : | x | < = [0 , ). Thus, openness and closedness depend on the underyling metric space as well as on the set. Example: In any metric space ( X, d ) both and X are open, and both and X are closed. To see that is open, note that the 1 statement x > B ( x ) X is vacuously true since thare arent any x . To see that X is open, note that since B ( x ) is by definition { z X : d ( z, x ) < } , it is trivially contained in X . Since is open, X is closed; since X is open, is closed. Theorem 2 (4.2) Let ( X, d ) be a metric space. Then 1. and X are both open, and both closed. 2. The union of an arbitrary (finite, countable, or uncount- able) collection of open sets is open. 3. The intersection of a finite collection of open sets is open. Proof: 1. We have already done this. 2. Suppose { A } is a collection of open sets. x [ A x A > B ( x ) A [ A so A is open....
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204Lecture42006 - Economics 204 Lecture 4August 3, 2006...

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