204CorrectionstodelaFuenteTimeless

204CorrectionstodelaFuenteTimeless - Econ 204 Corrections...

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Econ 204 Corrections to de la Fuente Note that Item 10 was added 7/22/09 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 ) = . 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 ) = ; 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 ) = .” 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 0 a limit point of A . Then f has limit y 0 as x x 0 if and only if for every sequence { x n } that converges to x 0 in ( X, d ) with x n A for every n and x n = x 0 , the sequence { f ( x n ) } converges to y 0 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 0 X and thus, x 0 must be in the domain of f . The revised statement just given allows x 0 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 1
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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 ) .” 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 f n f m s < ε/ 3. Fix m > N . Then n > N f n f m s < ε/ 3, so for each x X , | f ( x ) f m ( x ) | = lim n →∞ | f n ( x ) f m ( x ) | ≤ ε/ 3. Since x is arbitrary, m > N f f m s ε/ 3 < ε , so f f m s 0, i.e. { f m } converges to f in the sup norm. To see that f is continuous, fix x 0 X ; let N be as in the previous bullet and take m = N + 1. Since f N +1 is continuous, there exists δ > 0 such that | x x 0 | < δ ⇒ | f N +1 ( x ) f N +1 ( x 0 ) | < ε/ 3.
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