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
icon
Related questions
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)
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
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,