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1. Consider the sets D = { // : n € J}, E = DU{0}. (a) Really quick: complete the definitions - (a) a set E is an open set if... (b) a set E is a closed set if... (c) a set E is a compact set if... (b) For each of D and E decide whether it is closed and whether it is open, and give reason. (c) Find an open cover of D that does not have a finite sub-cover. Prove it. (d) Suppose UE is an open cover of E. Describe precisely how you would determine a finite sub-cover. &EA (e) Using the definition, (a) Is D compact? (b) Is E compact?

1. Consider the sets D = { // : n € J}, E = DU{0}. (a) Really quick: complete the definitions - (a) a set E is an open set if... (b) a set E is a closed set if... (c) a set E is a compact set if... (b) For each of D and E decide whether it is closed and whether it is open, and give reason. (c) Find an open cover of D that does not have a finite sub-cover. Prove it. (d) Suppose UE is an open cover of E. Describe precisely how you would determine a finite sub-cover. &EA (e) Using the definition, (a) Is D compact? (b) Is E compact?

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. Consider the sets D = {: ne J}, E = DU {0}. (a) Really quick: complete the definitions - (a) a set E is an open set if... (b) a set E is a closed set if... (c) a set E is a compact set if... (b) For each of D and E decide whether it is closed and whether it is open, and give reason. (c) Find an open cover of D that does not have a finite sub-cover. Prove it. (d) Suppose U Ea is an open cover of E. Describe precisely how you would determine a finite sub-cover. αEA (e) Using the definition, (a) Is D compact? (b) Is E compact?
Transcribed Image Text:1. Consider the sets D = {: ne J}, E = DU {0}. (a) Really quick: complete the definitions - (a) a set E is an open set if... (b) a set E is a closed set if... (c) a set E is a compact set if... (b) For each of D and E decide whether it is closed and whether it is open, and give reason. (c) Find an open cover of D that does not have a finite sub-cover. Prove it. (d) Suppose U Ea is an open cover of E. Describe precisely how you would determine a finite sub-cover. αEA (e) Using the definition, (a) Is D compact? (b) Is E compact?
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