4.1 Prove that in any metric space (S, d) every closed ball S,ro) is a closed set. 4.2 Let F be a closed set for k = 1, 2, ...,n in (S, d). Show that F is closed. %3D k=1 4.3 Give an example to show that the arbitrary intersection of open sets is not an open set.
4.1 Prove that in any metric space (S, d) every closed ball S,ro) is a closed set. 4.2 Let F be a closed set for k = 1, 2, ...,n in (S, d). Show that F is closed. %3D k=1 4.3 Give an example to show that the arbitrary intersection of open sets is not an open set.
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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![4.1 Prove that in any metric space (S, d) every closed ball S,ro] is a closed set.
4.2 Let F be a closed set for k 1, 2,... ,n in (S, d). Show that F, is closed.
k=1
4.3 Give an example to show that the arbitrary intersection of open sets is not an
open set.
4.4 Prove that if A is nowhere dense in discrete space (S, d) then A= 0.
4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else
contains some nonempty open set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d425c7-dc72-45ef-bec6-7e38b4e50bb4%2Fd5ffe317-87d1-4a50-941f-a1c6e81c4bc6%2F1liipmj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.1 Prove that in any metric space (S, d) every closed ball S,ro] is a closed set.
4.2 Let F be a closed set for k 1, 2,... ,n in (S, d). Show that F, is closed.
k=1
4.3 Give an example to show that the arbitrary intersection of open sets is not an
open set.
4.4 Prove that if A is nowhere dense in discrete space (S, d) then A= 0.
4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else
contains some nonempty open set.
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