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Use the following definition of compactness: KCR is compact if every open covering B of K has a finite subcovering CCB to show that (a,b] CR, a

Use the following definition of compactness: KCR is compact if every open covering B of K has a finite subcovering CCB to show that (a,b] CR, a

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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Please answer both questions 3a and b fully. Also please use the definition of compactness and not Heine Borel Theorem to show the proofs

3. Use the following definition of compactness: KCR is compact if every open covering B of K has a finite subcovering CC B to show that a. b. (a,b] CR, a <b is no compact. Hint: Consider B = {(a + , b+ 1) : n€ N}. Show that B is an open covering of (a, b], i.e. show rigorously that (a, b] CUB ВЕВ • Show that B has no finite subcovering CCB. K = {₁, T2, ..., n} CR is compact. Hint: Consider an arbitrary open covering B of K. Argue that for each r; € K, there is B; € B such that ; € B₁, for each i = 1,2,..., n. • Explain why is C = {B₁,..., B₁} a finite subcovering of B.
Transcribed Image Text:3. Use the following definition of compactness: KCR is compact if every open covering B of K has a finite subcovering CC B to show that a. b. (a,b] CR, a <b is no compact. Hint: Consider B = {(a + , b+ 1) : n€ N}. Show that B is an open covering of (a, b], i.e. show rigorously that (a, b] CUB ВЕВ • Show that B has no finite subcovering CCB. K = {₁, T2, ..., n} CR is compact. Hint: Consider an arbitrary open covering B of K. Argue that for each r; € K, there is B; € B such that ; € B₁, for each i = 1,2,..., n. • Explain why is C = {B₁,..., B₁} a finite subcovering of B.
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