Exercise 3. Are the following subsets of the real line with its usual topology (in which U CR is open if and only if for every x € U there is € > 0 with (x − ¤, x + €) ≤ U) open, closed, both or neither? Explain. (1) [1,3); (2) (1,3) U (5, ∞); (3) (-∞, ∞); (4) { ½ : n € N} U {0}; (5) The set Q of rationals.
Exercise 3. Are the following subsets of the real line with its usual topology (in which U CR is open if and only if for every x € U there is € > 0 with (x − ¤, x + €) ≤ U) open, closed, both or neither? Explain. (1) [1,3); (2) (1,3) U (5, ∞); (3) (-∞, ∞); (4) { ½ : n € N} U {0}; (5) The set Q of rationals.
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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![**Exercise 3.** Are the following subsets of the real line with its usual topology (in which \(U \subset \mathbb{R}\) is open if and only if for every \(x \in U\) there is \(\epsilon > 0\) with \((x - \epsilon, x + \epsilon) \subset U\)) open, closed, both or neither? Explain.
1. \([1, 3]\);
2. \((1, 3) \cup (5, \infty)\);
3. \((-\infty, \infty)\);
4. \(\left\{\frac{1}{n} : n \in \mathbb{N}\right\} \cup \{0\}\);
5. The set \(\mathbb{Q}\) of rationals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c557725-037f-4839-bada-f183764fca9b%2Fd404d533-b5fb-426d-94b8-f95d286e9f38%2Fv6tz7de_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 3.** Are the following subsets of the real line with its usual topology (in which \(U \subset \mathbb{R}\) is open if and only if for every \(x \in U\) there is \(\epsilon > 0\) with \((x - \epsilon, x + \epsilon) \subset U\)) open, closed, both or neither? Explain.
1. \([1, 3]\);
2. \((1, 3) \cup (5, \infty)\);
3. \((-\infty, \infty)\);
4. \(\left\{\frac{1}{n} : n \in \mathbb{N}\right\} \cup \{0\}\);
5. The set \(\mathbb{Q}\) of rationals.
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