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Exercise 1. Show that a subset K of the topological space X is compact (for the subspace topology on K) if and only if for every family (U₁)ier of open subsets of X with K CUier U₁, we can find 1,...,in EI such that K CU1U₁₁.

Exercise 1. Show that a subset K of the topological space X is compact (for the subspace topology on K) if and only if for every family (U₁)ier of open subsets of X with K CUier U₁, we can find 1,...,in EI such that K CU1U₁₁.

Advanced Engineering Mathematics
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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1. **Compactness** **Exercise 1.** Show that a subset \( K \) of the topological space \( X \) is compact (for the subspace topology on \( K \)) if and only if for every family \( (U_i)_{i \in I} \) of open subsets of \( X \) with \( K \subset \bigcup_{i \in I} U_i \), we can find \( i_1, \ldots, i_n \in I \) such that \( K \subset \bigcup_{j=1}^{n} U_{i_j} \).
Transcribed Image Text:1. **Compactness** **Exercise 1.** Show that a subset \( K \) of the topological space \( X \) is compact (for the subspace topology on \( K \)) if and only if for every family \( (U_i)_{i \in I} \) of open subsets of \( X \) with \( K \subset \bigcup_{i \in I} U_i \), we can find \( i_1, \ldots, i_n \in I \) such that \( K \subset \bigcup_{j=1}^{n} U_{i_j} \).
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