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Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is closed.

Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is closed.

Advanced Engineering Mathematics
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: Properties of Open and Closed Sets in Topological Spaces** **Problem Statement:** *Prove that in a topological space \( X \), if \( U \) is open and \( C \) is closed, then \( U - C \) is open and \( C - U \) is closed.* **Explanation:** In the context of topology, the problem involves demonstrating the properties of set operations under the conditions that one set is open and the other is closed. Here's a breakdown of the terms and proof structure: - **Topological Space \( X \):** A set with a collection of open subsets that satisfy specific properties such as union and intersection rules. - **Open Set \( U \):** A set within the topological space \( X \) that belongs to the topology. - **Closed Set \( C \):** A set whose complement relative to \( X \) is open. - **Set Difference \( U - C \):** Represents the elements in \( U \) that are not in \( C \). - **Set Difference \( C - U \):** Represents the elements in \( C \) that are not in \( U \). By understanding these gaps, we can ascertain: 1. **\( U - C \) is Open:** Since \( U \) is open and \( C \) is closed (hence, \( X - C \) is open), the set difference \( U - C = U \cap (X - C) \). Here, the intersection of two open sets \( U \) and \( X - C \) is open. 2. **\( C - U \) is Closed:** To prove \( C - U \) is closed, note that its complement relative to \( X \) is open. Specifically, the complement is \( X - (C - U) = X - C \cup U \), an open set because it is a union of the open set \( X - C \) with \( U \). This conceptual framework and step-by-step reasoning demonstrate foundational principles in topology concerning open and closed sets.
Transcribed Image Text:**Title: Properties of Open and Closed Sets in Topological Spaces** **Problem Statement:** *Prove that in a topological space \( X \), if \( U \) is open and \( C \) is closed, then \( U - C \) is open and \( C - U \) is closed.* **Explanation:** In the context of topology, the problem involves demonstrating the properties of set operations under the conditions that one set is open and the other is closed. Here's a breakdown of the terms and proof structure: - **Topological Space \( X \):** A set with a collection of open subsets that satisfy specific properties such as union and intersection rules. - **Open Set \( U \):** A set within the topological space \( X \) that belongs to the topology. - **Closed Set \( C \):** A set whose complement relative to \( X \) is open. - **Set Difference \( U - C \):** Represents the elements in \( U \) that are not in \( C \). - **Set Difference \( C - U \):** Represents the elements in \( C \) that are not in \( U \). By understanding these gaps, we can ascertain: 1. **\( U - C \) is Open:** Since \( U \) is open and \( C \) is closed (hence, \( X - C \) is open), the set difference \( U - C = U \cap (X - C) \). Here, the intersection of two open sets \( U \) and \( X - C \) is open. 2. **\( C - U \) is Closed:** To prove \( C - U \) is closed, note that its complement relative to \( X \) is open. Specifically, the complement is \( X - (C - U) = X - C \cup U \), an open set because it is a union of the open set \( X - C \) with \( U \). This conceptual framework and step-by-step reasoning demonstrate foundational principles in topology concerning open and closed sets.
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