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2. Prove that the set Z of integers with restriction of usual metric on R is a complete metric space, yet it is the union of countably many singletons. Explain why this does not contradict the Baire category theorem?

2. Prove that the set Z of integers with restriction of usual metric on R is a complete metric space, yet it is the union of countably many singletons. Explain why this does not contradict the Baire category theorem?

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.4: Binary Operations
Problem 12E
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1. Prove each of the following: (i) The set R of rational numbers is first category in the set of Reals (ii) The set of rational numbers cannot be written as Q = EN Un where {Un ne N} is a sequence of open subsets of R, the set of all Reals. 2. Prove that the set 2 of integers with restriction of usual metric on R is a complete metric space, yet it is the union of countably many singletons. Explain why this does not contradict the Baire category theorem? 3. Suppose that {nk ke N} is a strictly increasing sequence of natural numbers and {xn n E N} is a sequence in a metric space X. Then we say the sequence {nk N} is a subsequence of {n}. Suppose that {n E N} is a Cauchy sequence in the metric space (X, d) which has a convergent subsequence. Prove that {n} converges. 4. Suppose that X, Y and Z are metric spaces and F X Y and g Y Z are continuous functions. prove that go f X→ Z is continuous. : 1
Transcribed Image Text:1. Prove each of the following: (i) The set R of rational numbers is first category in the set of Reals (ii) The set of rational numbers cannot be written as Q = EN Un where {Un ne N} is a sequence of open subsets of R, the set of all Reals. 2. Prove that the set 2 of integers with restriction of usual metric on R is a complete metric space, yet it is the union of countably many singletons. Explain why this does not contradict the Baire category theorem? 3. Suppose that {nk ke N} is a strictly increasing sequence of natural numbers and {xn n E N} is a sequence in a metric space X. Then we say the sequence {nk N} is a subsequence of {n}. Suppose that {n E N} is a Cauchy sequence in the metric space (X, d) which has a convergent subsequence. Prove that {n} converges. 4. Suppose that X, Y and Z are metric spaces and F X Y and g Y Z are continuous functions. prove that go f X→ Z is continuous. : 1
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