2) If we assume that X = UiENF, with each F, a closed subset of the topological space X, and f: X→Y is a map such that every restriction fF: F→Y,ie N, is continuous, does it follow that f is automatically continuous? [If true prove it, if false give an example]
2) If we assume that X = UiENF, with each F, a closed subset of the topological space X, and f: X→Y is a map such that every restriction fF: F→Y,ie N, is continuous, does it follow that f is automatically continuous? [If true prove it, if false give an example]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
1.2
![2) If we assume that \( X = \bigcup_{i \in \mathbb{N}} F_i \), with each \( F_i \) a closed subset of the topological space \( X \), and \( f : X \rightarrow Y \) is a map such that every restriction \( f|F_i : F_i \rightarrow Y, i \in \mathbb{N} \), is continuous, does it follow that \( f \) is automatically continuous? [If true prove it, if false give an example]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2Fab1f6b89-eba2-4806-a80a-dc3b42e334cf%2Fx7gm7ni_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2) If we assume that \( X = \bigcup_{i \in \mathbb{N}} F_i \), with each \( F_i \) a closed subset of the topological space \( X \), and \( f : X \rightarrow Y \) is a map such that every restriction \( f|F_i : F_i \rightarrow Y, i \in \mathbb{N} \), is continuous, does it follow that \( f \) is automatically continuous? [If true prove it, if false give an example]
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