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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?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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