Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset of R" and F is a closed subset of K, then F is also compact.
Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset of R" and F is a closed subset of K, then F is also 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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(Second picture is the definition of compactness)
![### Compactness in Topology
(b) We say that \( K \) is **compact** if every open cover of \( K \) has a finite subcover; that is, if \( \mathcal{J} \) is an open cover of \( K \), we can find finitely many sets in \( \mathcal{J} \), say \( G_1, G_2, \ldots, G_n \) such that
\[ K \subseteq \bigcup_{j=1}^{n} G_j. \]
_DEFINED TERMS_
- **Open cover**: A collection of open sets whose union includes the space \( K \).
- **Finite subcover**: A finite number of sets from the original open cover that still covers \( K \).
This concept is fundamental in topology and essential for understanding various properties and behaviors of topological spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2Fb2e9be4c-66b3-4e3c-b292-705ea859545e%2Fzzgo7dn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Compactness in Topology
(b) We say that \( K \) is **compact** if every open cover of \( K \) has a finite subcover; that is, if \( \mathcal{J} \) is an open cover of \( K \), we can find finitely many sets in \( \mathcal{J} \), say \( G_1, G_2, \ldots, G_n \) such that
\[ K \subseteq \bigcup_{j=1}^{n} G_j. \]
_DEFINED TERMS_
- **Open cover**: A collection of open sets whose union includes the space \( K \).
- **Finite subcover**: A finite number of sets from the original open cover that still covers \( K \).
This concept is fundamental in topology and essential for understanding various properties and behaviors of topological spaces.
Transcribed Image Text:### Question 2
Prove directly (i.e. from the definition of compactness) that if \( K \) is a compact subset of \( \mathbb{R}^p \) and \( F \) is a closed subset of \( K \), then \( F \) is also compact.
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