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3) Suppose X = Uier Ui, with each U₁ an open subset of X. If for each i E I, the subset U₁ is a Baire space for the induced topology on the subset U, C X, show that X itself is a Baire space. Assumption In the remaining part of this exercise, we assume that X is a space such that every point x EX has a neighborhood V in X which is a Baire space for the topology induced on the subset V CX.

3) Suppose X = Uier Ui, with each U₁ an open subset of X. If for each i E I, the subset U₁ is a Baire space for the induced topology on the subset U, C X, show that X itself is a Baire space. Assumption In the remaining part of this exercise, we assume that X is a space such that every point x EX has a neighborhood V in X which is a Baire space for the topology induced on the subset V CX.

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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Exercise 2 Part 3
**Exercise 2**: Let \( X \) be a topological space. 3) Suppose \( X = \bigcup_{i \in I} U_i \), with each \( U_i \) an open subset of \( X \). If for each \( i \in I \), the subset \( U_i \) is a Baire space for the induced topology on the subset \( U_i \subset X \), show that \( X \) itself is a Baire space. **Assumption**: In the remaining part of this exercise, we assume that \( X \) is a space such that every point \( x \in X \) has a neighborhood \( V_x \) in \( X \) which is a Baire space for the topology induced on the subset \( V_x \subset X \).
Transcribed Image Text:**Exercise 2**: Let \( X \) be a topological space. 3) Suppose \( X = \bigcup_{i \in I} U_i \), with each \( U_i \) an open subset of \( X \). If for each \( i \in I \), the subset \( U_i \) is a Baire space for the induced topology on the subset \( U_i \subset X \), show that \( X \) itself is a Baire space. **Assumption**: In the remaining part of this exercise, we assume that \( X \) is a space such that every point \( x \in X \) has a neighborhood \( V_x \) in \( X \) which is a Baire space for the topology induced on the subset \( V_x \subset X \).
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