Using Heine-Borel: **Compact Sets in Real Analysis***Let* \(\mathcal{F} = \{ F_\alpha : \alpha \in A \}\) *be a family of compact subsets of* \(\mathbb{R}\). *Prove* \(\bigcap_{\alpha \in A} F_\alpha\) *is compact.***Explanation:**In this statement, we are dealing with the concept of compactness in the context of real analysis. The family \(\mathcal{F}\) consists of sets \(F_\alpha\), where each \(F_\alpha\) is a compact subset of the real numbers \(\mathbb{R}\). Compactness in \(\mathbb{R}\) typically involves sets that are both closed and bounded, according to the Heine-Borel theorem. The goal is to show that the intersection of these compact sets, represented by \(\bigcap_{\alpha \in A} F_\alpha\), is also compact.The process generally involves demonstrating that this intersection remains both closed and bounded, using properties of compact sets and understanding that the intersection of any collection of closed sets is closed, and a finite intersection of bounded sets is bounded. This proof is fundamental in topology and real analysis, emphasizing the stability of compactness under set intersections.

Answered: Let F= {Fa: a € A} be a family of…

Math

Advanced Math

Let F= {Fa: a € A} be a family of compact subsets of R. Prove a€A Fa is compact.

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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Question

Using Heine-Borel:

**Compact Sets in Real Analysis**

*Let* \(\mathcal{F} = \{ F_\alpha : \alpha \in A \}\) *be a family of compact subsets of* \(\mathbb{R}\). *Prove* \(\bigcap_{\alpha \in A} F_\alpha\) *is compact.*

**Explanation:**

In this statement, we are dealing with the concept of compactness in the context of real analysis. The family \(\mathcal{F}\) consists of sets \(F_\alpha\), where each \(F_\alpha\) is a compact subset of the real numbers \(\mathbb{R}\). Compactness in \(\mathbb{R}\) typically involves sets that are both closed and bounded, according to the Heine-Borel theorem. The goal is to show that the intersection of these compact sets, represented by \(\bigcap_{\alpha \in A} F_\alpha\), is also compact.

The process generally involves demonstrating that this intersection remains both closed and bounded, using properties of compact sets and understanding that the intersection of any collection of closed sets is closed, and a finite intersection of bounded sets is bounded. This proof is fundamental in topology and real analysis, emphasizing the stability of compactness under set intersections.

Expert Solution

Step 1: Explanation

Heine-Borel Theorem : If a set S of real numbers is closed and bounded, then the set S is compact. That is, if a set S of real numbers is closed and bounded, then every open cover of the set S has a finite subcover.

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