204Lecture42006

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

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Unformatted text preview: Economics 204 Lecture 4–August 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 ∃ ε> 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 } > 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 − | < ε = [0 , ε ). 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 ∈∅ ∃ ε> B ε ( x ) ⊆ X is vacuously true since thare 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. 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 λ } λ ∈ Λ is a collection of open sets. x ∈ [ λ ∈ Λ A λ ⇒ ∃ λ ∈ Λ x ∈ A λ ⇒ ∃ ε> B ε ( x ) ⊆ A λ ⊆ [ λ ∈ Λ A λ so ∪ λ ∈ Λ A λ is open....
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This note was uploaded on 03/06/2009 for the course ECON 0204 taught by Professor Staff during the Summer '08 term at Berkeley.

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

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