204CorrectionstodelaFuente0806 - Econ 204–Summer/Fall...

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Unformatted text preview: Econ 204–Summer/Fall 2006 Corrections to de la Fuente 1. On page 23, de la Fuente presents two definitions of correspondence. In the second definition, de la Fuente requires that for all x ∈ X , Ψ( x ) 6 = ∅ . The first definition simply says that Ψ is a function from X to 2 Y , the collection of all subsets of Y , and appears to believe that this implies that Ψ( x ) 6 = ∅ ; since ∅ ∈ 2 Y , this belief is not correct. In the first definition, he should have said “a correspondence is a function from X to 2 Y such that for all x ∈ X , Ψ( x ) 6 = ∅ .” 2. Theorem 5.2, page 64, should read as follows: Theorem 1 (5.2’) Let ( X, d ) and ( Y, ρ ) be two metric spaces, A ⊆ X , f : A → Y , and x a limit point of A . Then f has limit y as x → x if and only if for every sequence { x n } that converges to x in ( X, d ) with x n ∈ A for every n and x n 6 = x , the sequence { f ( x n ) } converges to y in ( Y, ρ ) . Comment: As stated in de la Fuente, the metric space ( X, d ) is the ambient space, so every limit point of X must be an element of X ; there is nothing outside of X to which a sequence in X can converge. Thus, as stated in de la Fuente, we must have x ∈ X and thus, x must be in the domain of f . The revised statement just given allows x to lie outside the domain of f . 3. De la Fuente uses a weaker definition of homeomorphism (Definition 6.20, page 74) than most texts; usually, a homeomorphism is required to be a surjection. For example, the injection map I : [0 , 1] → R defined by I ( x ) = x would not be called a homeomorphism in most texts because it is not onto, but it is a homeomorphism under de la Fuente’s weaker definition. This creates trouble in Theorem 6.21(ii). [0 , 1] is an open set in the metric space [0 , 1], but its image I ([0 , 1]) = [0 , 1] is not an open set in R , so Theorem 6.21 is false as stated. Theorem 6.21 is true if we assume that f : ( X, d ) → ( Y, ρ ) is one-to-one and onto , or if we replace the phrase “its image f ( A X ) is open in ( Y, ρ )” in part (ii) with “its image f ( A X ) is open in f ( X ) , ρ | f ( X ) .” 1 4. The proof of Theorem 7.12 is a bit disorganized and hard to follow. In the second bullet on page 84, de la Fuente assumes that { f n } converges to f in the sup norm, but this is not proven until the third bullet. Since the third bullet does not use the continuity of the limit function f , we could simply switch the second and third bullets to get a correct (but awkward) proof. Here is a better alternative to the second and third bullets: • Fix ε > 0. Since the sequence { f n } is Cauchy in the sup norm, there exists N such n, m > N ⇒ k f n − f m k s < ε/ 3. Fix m > N ....
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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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204CorrectionstodelaFuente0806 - Econ 204–Summer/Fall...

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