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Set Berived of every set is a closed set Lef A' is the set Cimit of we have to show that A' is closed. . Let XE € (A¹) ² be an abitrany point neighbourhood So 7 Contain limit pänt of A this means points of a Set A of X, NEGx) Sit NEC NE (₂) CAC Since every point of NE (x) is an interior point NEGx) not Contain any point of A C Calc

Set Berived of every set is a closed set Lef A' is the set Cimit of we have to show that A' is closed. . Let XE € (A¹) ² be an abitrany point neighbourhood So 7 Contain limit pänt of A this means points of a Set A of X, NEGx) Sit NEC NE (₂) CAC Since every point of NE (x) is an interior point NEGx) not Contain any point of A C Calc

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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Basic Topology Please check my solution to Q7, I was told there is a counter example to that solution?? Kindly show how and what is the correct solution
Set Derived of every # Lef A' So 7 Sef Contain this the set Cimit of points of we have to show that A' is closed. XE (A¹) C be Let an is a closed set (Q denote the set of n a Set A. abitrany point neighbourhood of X, NEGx) Sot NE(x) not limit pant of A NE (x) CAC means Since every point of NE (x) is an interior point of NeC) NE (x) not contain any point of A SO, NE (x) ≤ (A¹) C then (A)C is open 23 So A is closed. 9. Consider the set of real number IR with Standard on R on R. (usual topology fopology Lef A = Q rational number)
Transcribed Image Text:Set Derived of every # Lef A' So 7 Sef Contain this the set Cimit of points of we have to show that A' is closed. XE (A¹) C be Let an is a closed set (Q denote the set of n a Set A. abitrany point neighbourhood of X, NEGx) Sot NE(x) not limit pant of A NE (x) CAC means Since every point of NE (x) is an interior point of NeC) NE (x) not contain any point of A SO, NE (x) ≤ (A¹) C then (A)C is open 23 So A is closed. 9. Consider the set of real number IR with Standard on R on R. (usual topology fopology Lef A = Q rational number)
For each of the following, if the statement about a topological space is always true, prove it; otherwise, give a counterexample. 7. The derived set of every set is a closed set. 8. The boundary of every set is a closed set. 9. The boundary of every set has empty interior. 10. Every nonempty closed set with empty interior is the boundary of some set. 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with the closure of its interior. " 13. The interior of a set coincides with the interior of its closure.. 14. Every set that does not meet its boundary is open. 2 Base for a Topology next introduce the idea of a base for a topology, a concept that yields a ain economy of thought and effort in defining a topology for a set. Definition Lot (YT) be a tonological space. A subset B of T such that every element
Transcribed Image Text:For each of the following, if the statement about a topological space is always true, prove it; otherwise, give a counterexample. 7. The derived set of every set is a closed set. 8. The boundary of every set is a closed set. 9. The boundary of every set has empty interior. 10. Every nonempty closed set with empty interior is the boundary of some set. 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with the closure of its interior. " 13. The interior of a set coincides with the interior of its closure.. 14. Every set that does not meet its boundary is open. 2 Base for a Topology next introduce the idea of a base for a topology, a concept that yields a ain economy of thought and effort in defining a topology for a set. Definition Lot (YT) be a tonological space. A subset B of T such that every element
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