**Title: Constructing a Family of Sets to Demonstrate Non-Compactness in \( \mathbb{R} \)****Introduction:**Compactness is a fundamental concept in topology and analysis. A set is compact if every open cover has a finite subcover. This property does not hold for certain spaces, and the real number line \( \mathbb{R} \) is a classic example. Here's how you can construct a family of sets to demonstrate that \( \mathbb{R} \) is not compact.**Understanding the Concept:**To show that \( \mathbb{R} \) is not compact, consider constructing an open cover of \( \mathbb{R} \) that has no finite subcover. This construction involves creating a collection of open intervals that cover all real numbers but cannot be reduced to a finite number of intervals still covering all of \( \mathbb{R} \).**Construction Example:**Consider the family of open intervals:\[ \mathcal{F} = \{ (n, n+2) \mid n \in \mathbb{Z} \} \]where \( \mathbb{Z} \) is the set of all integers. Each interval \( (n, n+2) \) covers real numbers just beyond each integer point.- **Union of the family:** The union of these sets, \( \bigcup_{n \in \mathbb{Z}} (n, n+2) \), covers the entire real line, as every real number \( x \in \mathbb{R} \) lies within some interval \( (n, n+2) \).- **Non-existence of finite subcover:** No finite subset of this family can cover \( \mathbb{R} \) because removing intervals will leave gaps corresponding to numbers not covered by any remaining intervals.**Conclusion:**This example illustrates that \( \mathbb{R} \) is not compact because it does not satisfy the finite subcover condition. Understanding such properties is crucial for deeper insights into topology and the structure of real numbers.

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Answered: Construct a family of sets that…

Construct a family of sets that demonstrates R is not compact

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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