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► Exercise 3.7. Suppose X is a topological space and UC SCX. (a) Show that the closure of U in S is equal to Un S. (b) Show that the interior of U in S contains Int Un S; give an example to show that they might not be equal.

► Exercise 3.7. Suppose X is a topological space and UC SCX. (a) Show that the closure of U in S is equal to Un S. (b) Show that the interior of U in S contains Int Un S; give an example to show that they might not be equal.

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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**Exercise 3.7** Suppose \( X \) is a topological space and \( U \subseteq S \subseteq X \). (a) Show that the closure of \( U \) in \( S \) is equal to \( \overline{U} \cap S \). (b) Show that the interior of \( U \) in \( S \) contains \( \text{Int} U \cap S \); give an example to show that they might not be equal. *Note: This exercise deals with understanding the concepts of closure and interior within the context of subsets in topology. It explores the relationship and differences between these notions when applied in a subspace.*
Transcribed Image Text:**Exercise 3.7** Suppose \( X \) is a topological space and \( U \subseteq S \subseteq X \). (a) Show that the closure of \( U \) in \( S \) is equal to \( \overline{U} \cap S \). (b) Show that the interior of \( U \) in \( S \) contains \( \text{Int} U \cap S \); give an example to show that they might not be equal. *Note: This exercise deals with understanding the concepts of closure and interior within the context of subsets in topology. It explores the relationship and differences between these notions when applied in a subspace.*
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