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Exercise 5. a) Suppose X is a Hausdorff space, and let Cn+1 C Cn CX be a non-increasing sequence of compact connected subsets. Show that the intersection neN Cn is connected. b) Find a non-increasing sequence of closed connected subsets Fn+1 C F C R² such that neN Fn is not connected. c) Show that any non-increasing sequence of connected subsets In+1 C In CR has a connected intersection EN In.

Exercise 5. a) Suppose X is a Hausdorff space, and let Cn+1 C Cn CX be a non-increasing sequence of compact connected subsets. Show that the intersection neN Cn is connected. b) Find a non-increasing sequence of closed connected subsets Fn+1 C F C R² such that neN Fn is not connected. c) Show that any non-increasing sequence of connected subsets In+1 C In CR has a connected intersection EN In.

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 5 Need part a, b, and c
5b) The sequence of F_n should be non-increasing (F_{n+1} included in F_n as written) 5c) The sequence of I_n should be non-increasing (I_{n+1} included in I_n as written)
Transcribed Image Text:5b) The sequence of F_n should be non-increasing (F_{n+1} included in F_n as written) 5c) The sequence of I_n should be non-increasing (I_{n+1} included in I_n as written)
Exercise 5. a) Suppose X is a Hausdorff space, and let Cn+1 C Cn C X be a non-increasing sequence of compact connected subsets. Show that the intersection EN Cn is connected. b) Find a non-increasing sequence of closed connected subsets Fn+1 C F C R² such that nen Fn is not connected. c) Show that any non-increasing sequence of connected subsets In+1 C In CR has a connected intersection EN In.
Transcribed Image Text:Exercise 5. a) Suppose X is a Hausdorff space, and let Cn+1 C Cn C X be a non-increasing sequence of compact connected subsets. Show that the intersection EN Cn is connected. b) Find a non-increasing sequence of closed connected subsets Fn+1 C F C R² such that nen Fn is not connected. c) Show that any non-increasing sequence of connected subsets In+1 C In CR has a connected intersection EN In.
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