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1.92 Referring to the definition in Exercise 1.91, answer the following questions. a) Prove that every closed finite interval [a, b] is a closed set. b) Give an example of subset E CR for which E is neither open nor closed. Justify your example. c) Give an example of a set SCR that is both open and closed.

1.92 Referring to the definition in Exercise 1.91, answer the following questions. a) Prove that every closed finite interval [a, b] is a closed set. b) Give an example of subset E CR for which E is neither open nor closed. Justify your example. c) Give an example of a set SCR that is both open and closed.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 22E
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1.92 Referring to the definition in Exercise 1.91, answer the following questions. a) Prove that every closed finite interval [a, b] is a closed set. b) Give an example of subset ECR for which E is neither open nor closed. Justify your example. c) Give an example of a set SCR that is both open and closed. For Reference 1.91 A subset E CR is called closed if and only if its complement R \ E is open. (For example, R itself is a closed set since R \ R = Ø is an open set.) Prove that a closed set E that is also dense in R must be all of R. (Hint: Suppose the claim were false, so that R \ E is a nonempty open set. Deduce a contradiction.)
Transcribed Image Text:1.92 Referring to the definition in Exercise 1.91, answer the following questions. a) Prove that every closed finite interval [a, b] is a closed set. b) Give an example of subset ECR for which E is neither open nor closed. Justify your example. c) Give an example of a set SCR that is both open and closed. For Reference 1.91 A subset E CR is called closed if and only if its complement R \ E is open. (For example, R itself is a closed set since R \ R = Ø is an open set.) Prove that a closed set E that is also dense in R must be all of R. (Hint: Suppose the claim were false, so that R \ E is a nonempty open set. Deduce a contradiction.)
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