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
Question

Part b only

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.
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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