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. aEA (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. aEA (e) Using the definition, (a) Is D compact? (b) Is E compact?
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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Can you do C,D,E please? And can you prove C thanks
Transcribed Image Text:1. Consider the sets \( D = \left\{ \frac{1}{n} : n \in J \right\}, E = D \cup \{0\} \).
(a) Really quick: complete the definitions -
\(\quad\)(a) a set \( E \) is an open set if ...
\(\quad\)(b) a set \( E \) is a closed set if ...
\(\quad\)(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 \( \bigcup_{\alpha \in A} E_\alpha \) is an open cover of \( E \). Describe precisely how you would determine a finite sub-cover.
(e) Using the definition, (a) Is \( D \) compact? (b) Is \( E \) compact?
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