Math 301 Problem Set 7 Solutions

# Math 301 Problem Set 7 Solutions - Math 301/ENAS 513 HW 7...

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Unformatted text preview: Math 301/ENAS 513: HW 7 Soln of Selected Problems Triet Le 1) Page 78: 5: Solution 1 . Let A ⊂ R ⊂ C with the usual metric d ( x, y ) = | x- y | . Pick any x ∈ A , then for any > 0, x + i / 2 ∈ N ( x ). Hence, N ( x ) is not a subset of A , for all > 0. And so x is not an interior point of A . 2) Page 78: 8: Determine the closure of S = { ( x 1 , sin(1 /x 1 )) : x 1 6 = 0 } ⊂ R 2 with the Euclidean metric. Solution 2 . Claim B = cl ( S ) = S ∪ { (0 , b ) :- 1 ≤ b ≤ 1 } . First we’ll show that any point outside of B is not a limit point of S . Pick any point x = ( x 1 , x 2 ) ∈ R 2 such that x 1 < 0. Then any y = ( y 1 , y 2 ) ∈ N | x | / 2 ( x ) has y 1 < 0. This N | x | / 2 ( x ) ∩ S = ∅ . Thus x with x 1 < 0 is not a limit point of S . A similar argument shows that x = ( x 1 , x 2 ) ∈ R 2 with x 1 = 1 and | x 2 | > 1 is not a limit point of S . Now for each x = ( x 1 , x 2 ) ∈ R 2 such that x 1 > 0 and x 2 6 = sin(1 /x 1 ). Then for = x 1 / 2, the set S x 1 = { ( a, sin(1 /a )) ∈ R 2 : a ∈ [ x 1- , x 2 + ] } is closed. This implies x ∈ ( S x 1 ) c ∩ { y = ( y 1 , y 2 ) ∈ R 2 : y 1 ∈ ( x 1- , x 1 + ) } , which is open. Hence, there exists 0 < δ <...
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Math 301 Problem Set 7 Solutions - Math 301/ENAS 513 HW 7...

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