1.91 A subset E C R 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.)
1.91 A subset E C R 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.)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.1: The Field Of Real Numbers
Problem 3TFE: Label each of the following statements as either true or false. The least upper bound of a nonempty...
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