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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fb2ed72f6-4370-4cf1-b064-16ea823f40de%2Fho2lqri_processed.jpeg&w=3840&q=75)
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fb2ed72f6-4370-4cf1-b064-16ea823f40de%2Fcysyzn_processed.jpeg&w=3840&q=75)
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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