Use the following definition of compactness: KCR is compact if every open covering B of K has a finite subcovering CC B to show that If A, BCR are compact then AUB is compact. Hint: Consider an arbitrary open covering C of AUB. • Show that CA = {CEC: CnA0} and CB = {C EC: CnB 0} are open coverings of A and B respectively. • Use the definition of compactness for A and B to conclude that CA and CB have finite subcov- erings DA CCA and DB CCB respectively. • Use DA and DÅ to show that C has a finite subcovering DCC.
Use the following definition of compactness: KCR is compact if every open covering B of K has a finite subcovering CC B to show that If A, BCR are compact then AUB is compact. Hint: Consider an arbitrary open covering C of AUB. • Show that CA = {CEC: CnA0} and CB = {C EC: CnB 0} are open coverings of A and B respectively. • Use the definition of compactness for A and B to conclude that CA and CB have finite subcov- erings DA CCA and DB CCB respectively. • Use DA and DÅ to show that C has a finite subcovering DCC.
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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Please answer the question fully and please include the picture. Also please use the definition of compactness and not Heine Borel Theorem to show the proof
Transcribed Image Text:### Understanding Compactness in Real Analysis
**Definition of Compactness:**
For a subset \( K \subseteq \mathbb{R} \), \( K \) is compact if every open covering \( B \) of \( K \) has a finite subcovering \( C \subseteq B \).
**Problem Statement:**
Prove that if \( A, B \subseteq \mathbb{R} \) are compact, then \( A \cup B \) is compact.
**Hint:** Consider an arbitrary open covering \( C \) of \( A \cup B \).
**Steps to Show the Compactness of \( A \cup B \):**
1. **Open Coverings for \( A \) and \( B \):**
- Prove that \( C_A = \{ C \in C : C \cap A \neq \emptyset \} \) and \( C_B = \{ C \in C : C \cap B \neq \emptyset \} \) are open coverings of \( A \) and \( B \) respectively.
2. **Finite Subcoverings for \( A \) and \( B \):**
- Apply the definition of compactness to \( A \) and \( B \) to find finite subcoverings \( D_A \subseteq C_A \) and \( D_B \subseteq C_B \) respectively.
3. **Finite Subcovering for \( A \cup B \):**
- Use \( D_A \) and \( D_B \) to demonstrate that \( C \) has a finite subcovering \( D \subseteq C \).
This exercise involves understanding the intersection properties of open sets and how they relate to the concept of compactness. By showing the existence of finite subcoverings, you establish that the union of compact sets is also compact, reinforcing the fundamental principles of real analysis.
**Note:** There are no graphs or diagrams in the provided content.
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