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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F49b11e7e-c28d-4a3b-ad73-ea233c9264eb%2Ffyu7xud_processed.png&w=3840&q=75)
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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