Math 301 Problem Set 7 Solutions

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 well 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 < <...
View Full Document

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online