204Lecture42008Web - Economics 204 Lecture 4Thursday, July...

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Economics 204 Lecture 4–Thursday, July 31, 2008 Section 2.4, Open and Closed Sets Defnition 1 Let ( X, d ) be a metric space. A set A X is open if x A ε> 0 B ε ( x ) A Aset C X is closed if X \ C is open. Example: ( a, b )i sopenintheme t r icspace E 1 ( R with the usual Euclidean metric). Given x ( a, b ), a<x<b .L e t ε =m in { x a, b x } > 0 Then y B ε ( x ) y ( x ε, x + ε ) ( x ( x a ) ,x +( b x )) =( a, b ) so B ε ( x ) ( a, b ), so ( a, b )isopen . Notice that ε depends on x ;inpar t icu lar , ε gets smaller as x nears the boundary of the set. Example: In E 1 ,[ a, b ]isc losed . 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 ε (0) = { x [0 , 1] : | x 0 | =[0 ). Thus, openness and closedness depend on the underyling metric space as well as on the set. Example: Most sets are neither open nor closed. For example, in E 1 ,[0 , 1] (2 , 3) is neither open nor closed. Example: An open set may consist of a single point. For example, 1
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if X = N and d ( m, n )= | m n | ,then B 1 / 2 (1) = { m N : | m 1 | < 1 / 2 } = { 1 } . 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 statement x ∈∅ ε> 0 B ε ( x ) ⊆∅ is vacuously true since thare aren’t any x ∈∅ .T oseetha t X is open, note that since B ε ( x ) is by deFnition { z X : d ( z, x ) } , it is trivially contained in X .S i n c e 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 (Fnite, countable, or uncountable) collection of open sets is open. 3. The intersection of a Fnite collection of open sets is open. Proof: 1. We have already done this. 2. Suppose { A λ } λ Λ is a collection of open sets. x [ λ Λ A λ ⇒∃ λ 0 Λ x A λ 0 ε> 0 B ε ( x ) A λ 0 [ λ Λ A λ so λ Λ A λ is open. 3. Suppose A 1 ,...,A n X are open sets. If x ∈∩ n i =1 A i x A 1 ,x A 2 ,...,x A n 2
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so ε 1 > 0 ,...,ε n > 0 B ε 1 ( x ) A 1 ,...,B ε n ( x ) A n Let ε =m in { ε 1 ,...,ε n } > 0 ( Aside: this is where we need the fact that we are taking a Fnite union. The inFmum of an inFnite set of positive numbers could be zero.
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204Lecture42008Web - Economics 204 Lecture 4Thursday, July...

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