204Lecture42009 - Economics 204 Lecture 4Thursday Revised...

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Economics 204 Lecture 4–Thursday, July 30, 2009 Revised 7/31/09, Revisions Indicated by ** and Sticky Notes 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 ε> 0 B ε ( 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 ε = min { 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 ) is open. Notice that ε depends on x ; in particular, ε 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 ε (0) = { x [0 , 1] : | x 0 | < ε = [0 , ε ). 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 1
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E 1 , [0 , 1] (2 , 3) is neither open nor closed. Example: An open set may consist of a single point. For example, 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 there aren’t any x ∈ ∅ . To see that X is open, note that since B ε ( x ) is by definition { z X : d ( z, x ) < ε } , it is trivially contained in X . Since is open, X is closed; since X is open, is closed. Example: Open balls are open sets. Suppose y B ε ( x ). Then d ( x, y ) < ε . Let δ = ε d ( x, y ) > 0. If d ( z, y ) < δ , then d ( z, x ) d ( z, y ) + d ( y, x ) < δ + d ( x, y ) = ε d ( x, y ) + d ( x, y ) = ε so B δ ( y ) B ( x ), so B ε ( x ) is open. 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
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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 , then x A 1 , x A 2 , . . . , x A n so ε 1 > 0 ,...,ε n > 0 B ε 1 ( x ) A 1 , . . . , B ε n ( x ) A n Let ε = min { ε 1 , . . . , ε n } > 0 ( Aside: this is where we need the fact that we are taking a finite **intersection. The infimum of an infinite set of
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