MATH527_HW2 - Serge Ballif MATH 527 Homework 2 September...

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Serge Ballif MATH 527 Homework 2 September 14, 2007 Problem 1. Consider a map h : R ω R ω , h ( x 1 ,x 2 ,x 3 ,... ) = ( x 1 , 4 x 2 , 9 x 3 ,... ) . (a) Is h continuous, when R ω is given by the product topology? (b) Is h continuous, when R ω is given by the box topology? (c) Is h continuous, when R ω is given by the uniform topology? (a) Yes, h is continuous. To show this, we show that the inverse image under h of a basis set in R ω is open in R ω . An arbitrary basis set in R ω has the form U = Q i =1 U i for intervals U i = ( a i ,b i ) where the two equations a i = -∞ and b i = hold except for finitely many values of i . The calculation h - 1 ( { ( y 1 ,y 2 ,...,y i ,... ) } ) = { ( y 1 , 1 4 y 2 ,..., 1 i 2 y i ,... ) } shows that h - 1 ( U ) = Q i =1 ( 1 i 2 a i , 1 i 2 b i ). This inverse image is also a product of open intervals of which only finitely many are not all of R . Hence, h - 1 ( U ) is another basis element of R ω . Thus the inverse image of a basis element is open under the map h . Therefore, h is continuous. (b) Yes, h is continuous. The inverse image under h of a basis set U = Q i =1 ( a i ,b i ) is the basis set (and hence an open set) h - 1 ( U ) = Q i =1 ( 1 i 2 a i , 1 i 2 b i ). Hence, h is continuous. (c) No, h is not continuous. To show h is not continuous, we will show that the inverse image of an open ball does not contain an open ball (and hence is not open). An basis element of R ω in the uniform topology is of the form B ¯ p ( x,± ) = [ δ<± Y i =1 ( x i - δ,x i + δ ) . The property
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This note was uploaded on 04/01/2008 for the course MATH 527 taught by Professor Rotman,regina during the Fall '07 term at Pennsylvania State University, University Park.

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MATH527_HW2 - Serge Ballif MATH 527 Homework 2 September...

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