Ans-10 - Answers Assignment#10 Exercises 1.4 1(a True If s int S then s does not belong to bd S(b False Let S =(0 1 Then bd S = cfw_0 1 which is not a

Ans-10 - Answers Assignment#10 Exercises 1.4 1(a True If s...

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Answers, Assignment #10 Exercises 1.4 1. (a) True. If s int S , then s does not belong to bd S . (b) False. Let S = (0 , 1). Then bd S = { 0 , 1 } which is not a subset of S . (c) False. S = [0 , 1], bd S = { 0 , 1 } , S = bd S . (d) True. Suppose s S . If s / int S , then every neighborhood of s must intersect S c which implies s bd S . (e) True. s bd S implies N S = and N S c = for every neighborhood N of s . t bd S c implies N S c = and N ( S c ) c = N S = for every neighborhood N of t . Therefore bd S = bd S c . (f) False. S = [0 , 1], bd S = { 0 , 1 } ⊂ S . 2. (a) True. Let z N ( x, ). Let δ = min {| z - x | , | z - ( x + ) | , | z - ( x - ) |} . Then N ( z, δ ) N ( x, ). (b) True. Theorem 10 (a). (c) False. n =1 1 n , 1 - 1 n = (0 , 1). (d) False. n =1 ( - 1 n , 1 + 1 n ) = [0 , 1]. (e) True. Let T be a collection of closed sets. Then [ T T ∈T ] c = T c t ∈T . T closed implies T c is open and the union of a collection of open sets is open. Therefore, [ T T ∈T ] c is open and T T ∈T is closed. (f) False. The set of real numbers is both open and closed.
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  • Fall '08
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  • Topology, Metric space, Sn, Closed set, General topology, ∩TT ∈T

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