Ans-10 - Answers, Assignment #10 Exercises 1.4 1. (a) True....

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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 .I f s/ int S , then every neighborhood of s must intersect S c which implies s bd S . (e) True. s S implies N S ± = and N S c ± = for every neighborhood N of s . t S c implies N S c ± = and N ( S c ) c = N S ± = for every neighborhood N of t . Therefore bd S S c . (f) False. S , 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 ² 1 n , 1 - 1 n ³ , 1). (d) False. ´ n =1 ( - 1 n , 1+ 1 n ) , 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. 3. (a) Closed. N c is open.
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Ans-10 - Answers, Assignment #10 Exercises 1.4 1. (a) True....

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