**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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Advanced Math

Construct a family of sets that demonstrates R is not compact

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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**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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