6. Every finite set in R is compact. Why?
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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![Example: In the Euclidean metric space (R, I-1), we have the following:
1.
The closed interval (a , b] is compact because it closed and bounded.
2.
The interval [0, 00) is not compact because it not bounded.
3.
The open interval (0, 1) is not compact because it is not closed and there is an open covering of the form
{(0, 1-):n = 2,3,.) of (0, 1) but has no finite subcovering.
4.
Z is not compact because it is not bounded.
R is not compact because there is an open cover of the form { (-n, n) : n € Z* } but it has no finite subcovering.
5.
6.
Every finite set in R is compact. Why?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F574ad348-5dcd-423f-b2b7-6ae35e09b860%2F45b7f068-a5c1-42fd-84db-e69415163e22%2F2m5pihs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example: In the Euclidean metric space (R, I-1), we have the following:
1.
The closed interval (a , b] is compact because it closed and bounded.
2.
The interval [0, 00) is not compact because it not bounded.
3.
The open interval (0, 1) is not compact because it is not closed and there is an open covering of the form
{(0, 1-):n = 2,3,.) of (0, 1) but has no finite subcovering.
4.
Z is not compact because it is not bounded.
R is not compact because there is an open cover of the form { (-n, n) : n € Z* } but it has no finite subcovering.
5.
6.
Every finite set in R is compact. Why?
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