(e) A = (-1,1)U{2} in the standard topology on R %3D (f) A = (-1,1) U {2} in the lower limit topology on R (g) A = {(x,0) E R² \x € R} in the standard topology on R².
(e) A = (-1,1)U{2} in the standard topology on R %3D (f) A = (-1,1) U {2} in the lower limit topology on R (g) A = {(x,0) E R² \x € R} in the standard topology on R².
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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Just do E F and G
![**Determine the interior, closure, and limit points of each of the following sets.**
1. **(a)** \( A = (0, 1] \) in the lower-limit topology on \(\mathbb{R}\).
2. **(b)** \( A = (0, 1] \) in the finite-complement topology on \(\mathbb{R}\).
3. **(c)** \( A = \{a, c\} \) in \( X = \{a, b, c\} \) with topology \( X = \{X, \emptyset, \{a\}, \{a, c\}\} \).
4. **(d)** \( A = \{b\} \) in \( X = \{a, b, c\} \) with topology \( X = \{X, \emptyset, \{a\}, \{a, b\}\} \).
5. **(e)** \( A = (-1,1) \cup \{2\} \) in the standard topology on \(\mathbb{R}\).
6. **(f)** \( A = (-1,1) \cup \{2\} \) in the lower limit topology on \(\mathbb{R}\).
7. **(g)** \( A = \{(x, 0) \in \mathbb{R}^2 \mid x \in \mathbb{R}\} \) in the standard topology on \(\mathbb{R}^2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93507587-8486-4bd1-9be9-7c1c5dc70b2a%2F89e3598c-7cd7-41e9-8339-274338bd246f%2F3a3bfno_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine the interior, closure, and limit points of each of the following sets.**
1. **(a)** \( A = (0, 1] \) in the lower-limit topology on \(\mathbb{R}\).
2. **(b)** \( A = (0, 1] \) in the finite-complement topology on \(\mathbb{R}\).
3. **(c)** \( A = \{a, c\} \) in \( X = \{a, b, c\} \) with topology \( X = \{X, \emptyset, \{a\}, \{a, c\}\} \).
4. **(d)** \( A = \{b\} \) in \( X = \{a, b, c\} \) with topology \( X = \{X, \emptyset, \{a\}, \{a, b\}\} \).
5. **(e)** \( A = (-1,1) \cup \{2\} \) in the standard topology on \(\mathbb{R}\).
6. **(f)** \( A = (-1,1) \cup \{2\} \) in the lower limit topology on \(\mathbb{R}\).
7. **(g)** \( A = \{(x, 0) \in \mathbb{R}^2 \mid x \in \mathbb{R}\} \) in the standard topology on \(\mathbb{R}^2\).
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