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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. PADDCL TIA. 3) Suppose X = Uier U₁, with each U₁ an open subset of X. If for each i € I, the subset Uį is a Baire space for the induced topology on the subset U₁ CX, 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 V₂ C X.
Transcribed Image Text:Exercise 2, Let X be a topological space. PADDCL TIA. 3) Suppose X = Uier U₁, with each U₁ an open subset of X. If for each i € I, the subset Uį is a Baire space for the induced topology on the subset U₁ CX, 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 V₂ C X.
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