2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite collection of open sets in a topological space (X, T), then |U¡ is оpen. i=1 (b) Give an example of a topological space and a collection of open sets in that topological space that shows that the infinite intersection of open sets need not be open.
2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite collection of open sets in a topological space (X, T), then |U¡ is оpen. i=1 (b) Give an example of a topological space and a collection of open sets in that topological space that shows that the infinite intersection of open sets need not be open.
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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Do a and b, I added what I did for A and the comment I got back
![2. (a) Use axiom (2) in the definition of a topology to show that if (U1, Uz....,Un} is a finite
collection of open sets in a topological space (X,T), then NU, is open.
(b) Give an example of a topological space and a collection of open sets in that topological
space that shows that the infinite intersection of open sets need not be open.
(a)
Proof -
iet su, Uz, Vs, Uy.. Un} is a finite collection of o pen sets in topological Space (x,7).
%3D
If xe
ñ U that is X€UL Such that 116h Since Ui is open, there is
Ai >0
Such that'e!
(x - Ai , X+ Ai) c i
NOw, A > min E{A,, Az Ag, Au
have lx-A, x+ A)C Aui. Then
A} >0. Hence cx-A, x +A) EUL. Then this argument won't work. you
Vi is open.
need to try math induction.
(b)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93507587-8486-4bd1-9be9-7c1c5dc70b2a%2F15d8646e-c700-4816-b2cb-395386bb6409%2Faums72_processed.png&w=3840&q=75)
Transcribed Image Text:2. (a) Use axiom (2) in the definition of a topology to show that if (U1, Uz....,Un} is a finite
collection of open sets in a topological space (X,T), then NU, is open.
(b) Give an example of a topological space and a collection of open sets in that topological
space that shows that the infinite intersection of open sets need not be open.
(a)
Proof -
iet su, Uz, Vs, Uy.. Un} is a finite collection of o pen sets in topological Space (x,7).
%3D
If xe
ñ U that is X€UL Such that 116h Since Ui is open, there is
Ai >0
Such that'e!
(x - Ai , X+ Ai) c i
NOw, A > min E{A,, Az Ag, Au
have lx-A, x+ A)C Aui. Then
A} >0. Hence cx-A, x +A) EUL. Then this argument won't work. you
Vi is open.
need to try math induction.
(b)
![2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite
collection of open sets in a topological space (X, T), then
|U¡ is
оpen.
i=1
(b) Give an example of a topological space and a collection of open sets in that topological
space that shows that the infinite intersection of open sets need not be open.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93507587-8486-4bd1-9be9-7c1c5dc70b2a%2F15d8646e-c700-4816-b2cb-395386bb6409%2Fhsnafjm_processed.png&w=3840&q=75)
Transcribed Image Text:2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite
collection of open sets in a topological space (X, T), then
|U¡ is
оpen.
i=1
(b) Give an example of a topological space and a collection of open sets in that topological
space that shows that the infinite intersection of open sets need not be open.
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