assignment5 - X can be written as the intersection of...

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MATH 255 ASSIGNMENT 5 This assignment is due in class on Monday, February 14 Problems Please justify carefully your answers. 1. [10 points] Let ( X, d X ) and ( Y, d Y ) be metric spaces and f : X Y . Prove that the following statements are equivalent. (1) f is continuous on X . (2) For any open set V Y , the set f - 1 ( V ) = { x X : f ( x ) V } , is open in X . 2. [25 points] Let X be a metric space and let E be a nonempty subset of X . Define d E ( x ) = inf { d ( x, e ) : e E } . (1) Show that d E ( x ) d ( x, y ) + d E ( y ). (2) Show that | d E ( x ) - d E ( y ) | ≤ d ( x, y ) and deduce that d E : X R is a continuous function. (3) If in addition E is a closed subset of X , show that d E ( x ) = 0 x E . (4) Deduce from (2) that for fixed t > 0, { x : d E ( x ) < t } is an open subset of X containing E . (5) Choosing t = 1 /n , n N in (4) show that every closed subset of
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Unformatted text preview: X can be written as the intersection of countably many open subsets of X . 3. [10 points] Let ( X,d ) be a complete metric space and f : X → X a continuous function such that for some integer N > 0, f N = f ◦ f ◦ ··· ◦ f | {z } N times , is a contraction on X . Prove that there exists unique x ∈ X such that f ( x ) = x . 4. [10 points] Consider Banach space C ([0 , 1]). Let F : C ([0 , 1]) → C ([0 , 1]) be a function defined by F ( f ) = sin( f ) . In other words, ( F ( f ))( t ) = sin( f ( t )) for t ∈ [0 , 1]. Prove that F is a continuous function. 1...
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