Find the closure of each of the following sets: (a) (3,5) U {6} (b) (-∞, 0) U (0, 1)
Find the closure of each of the following sets: (a) (3,5) U {6} (b) (-∞, 0) U (0, 1)
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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![Title: Understanding Set Closures
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**Topic: Set Closures**
**Objective:** Find the closure of each of the following sets.
---
**Problem Statement:**
(a) \( (3, 5) \cup \{6\} \)
(b) \( (-\infty, 0) \cup (0, 1) \)
**Discussion:**
The closure of a set in a topological space is essentially the smallest closed set that contains the original set. To determine the closure of these sets, consider the following:
- For interval \( (a, b) \), the closure is \([a, b]\), including endpoints.
- The union of intervals or sets should account for nearby limiting points to consider closure.
1. **Set A: \( (3, 5) \cup \{6\} \)**
- The interval \( (3, 5) \) does not include its endpoints, so its closure extends to \([3, 5]\).
- Since \( \{6\} \) is a discrete point, the closure remains \{6\}.
- Therefore, the closure of the entire set is the union of \([3, 5] \cup \{6\}\).
2. **Set B: \( (-\infty, 0) \cup (0, 1) \)**
- Consider the separate intervals:
- \( (-\infty, 0) \) is already closed towards \(-\infty\) but open at 0.
- \( (0, 1) \) does not include 0 or 1.
- Hence, when united, both intervals should account for the missing point 0, which makes the closure of the set \((-\infty, 1]\), adding the endpoint of 1 from the interval \( (0, 1) \).
**Graphical Representation (if needed):**
- **Diagram for (a):**
- A number line representation showing a filled segment from 3 to 5 and an individual point at 6.
- **Diagram for (b):**
- A number line displaying a filled curve extending from \(-\infty\) to 0 and a jump from 0 to 1.
These graphical interpretations illustrate how closures include boundary points of each interval to encompass all limit points associated with the given set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca338f80-b963-4565-b3ff-b52dc3ed88af%2F590aab9a-623e-4d62-8845-da83cb9bc87e%2Fndxramv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Understanding Set Closures
---
**Topic: Set Closures**
**Objective:** Find the closure of each of the following sets.
---
**Problem Statement:**
(a) \( (3, 5) \cup \{6\} \)
(b) \( (-\infty, 0) \cup (0, 1) \)
**Discussion:**
The closure of a set in a topological space is essentially the smallest closed set that contains the original set. To determine the closure of these sets, consider the following:
- For interval \( (a, b) \), the closure is \([a, b]\), including endpoints.
- The union of intervals or sets should account for nearby limiting points to consider closure.
1. **Set A: \( (3, 5) \cup \{6\} \)**
- The interval \( (3, 5) \) does not include its endpoints, so its closure extends to \([3, 5]\).
- Since \( \{6\} \) is a discrete point, the closure remains \{6\}.
- Therefore, the closure of the entire set is the union of \([3, 5] \cup \{6\}\).
2. **Set B: \( (-\infty, 0) \cup (0, 1) \)**
- Consider the separate intervals:
- \( (-\infty, 0) \) is already closed towards \(-\infty\) but open at 0.
- \( (0, 1) \) does not include 0 or 1.
- Hence, when united, both intervals should account for the missing point 0, which makes the closure of the set \((-\infty, 1]\), adding the endpoint of 1 from the interval \( (0, 1) \).
**Graphical Representation (if needed):**
- **Diagram for (a):**
- A number line representation showing a filled segment from 3 to 5 and an individual point at 6.
- **Diagram for (b):**
- A number line displaying a filled curve extending from \(-\infty\) to 0 and a jump from 0 to 1.
These graphical interpretations illustrate how closures include boundary points of each interval to encompass all limit points associated with the given set.
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