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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
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 \(\{U_1, U_2, \ldots, U_n\}\) is a finite collection of open sets in a topological space \((X, \mathcal{T})\), then \(\bigcap_{i=1}^n U_i\) 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 Let \(\{U_1, U_2, U_3, \ldots, U_n\}\) be a finite collection of open sets in topological space \((X, \mathcal{T})\). If \(x \in \bigcap U_i\) that is, \(x \in U_i\) such that \(1 \leq i \leq n\). Since \(U_i\) is open, there is \(A_i > 0\) such that \((x-A_i, x+A_i) \subseteq U_i\). Now, \(A = \min\{A_1, A_2, A_3, \ldots, A_n\} > 0\). Hence \((x-A, x+A) \subseteq U_i\). Then we have \((x-A, x+A) \subseteq \bigcap_{i=1}^n U_i\). Then \(\bigcap_{i=1}^n U_i\) is open. ### Diagram Explanation The diagram is a handwritten mathematical proof using set notation and logical steps to demonstrate the statement provided. It shows a logical flow of arguments involving open sets, subsets, and intervals using mathematical symbols. ### Note A comment in the image mentions that "this argument won't work. you need to try math induction," suggesting that there may be an issue with the logic applied or a more robust proof might require mathematical induction.
Transcribed Image Text:**2.** **(a)** Use axiom (2) in the definition of a topology to show that if \(\{U_1, U_2, \ldots, U_n\}\) is a finite collection of open sets in a topological space \((X, \mathcal{T})\), then \(\bigcap_{i=1}^n U_i\) 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 Let \(\{U_1, U_2, U_3, \ldots, U_n\}\) be a finite collection of open sets in topological space \((X, \mathcal{T})\). If \(x \in \bigcap U_i\) that is, \(x \in U_i\) such that \(1 \leq i \leq n\). Since \(U_i\) is open, there is \(A_i > 0\) such that \((x-A_i, x+A_i) \subseteq U_i\). Now, \(A = \min\{A_1, A_2, A_3, \ldots, A_n\} > 0\). Hence \((x-A, x+A) \subseteq U_i\). Then we have \((x-A, x+A) \subseteq \bigcap_{i=1}^n U_i\). Then \(\bigcap_{i=1}^n U_i\) is open. ### Diagram Explanation The diagram is a handwritten mathematical proof using set notation and logical steps to demonstrate the statement provided. It shows a logical flow of arguments involving open sets, subsets, and intervals using mathematical symbols. ### Note A comment in the image mentions that "this argument won't work. you need to try math induction," suggesting that there may be an issue with the logic applied or a more robust proof might require mathematical induction.
### Exercise 2 #### (a) Problem Statement Use axiom (2) in the definition of a topology to demonstrate that if \(\{U_1, U_2, \ldots, U_n\}\) is a finite collection of open sets in a topological space \((X, \mathcal{T})\), then the intersection \[ \bigcap_{i=1}^{n} U_i \] is open. #### (b) Problem Statement Provide an example of a topological space and a collection of open sets within that space to illustrate that the infinite intersection of open sets may not necessarily be open.
Transcribed Image Text:### Exercise 2 #### (a) Problem Statement Use axiom (2) in the definition of a topology to demonstrate that if \(\{U_1, U_2, \ldots, U_n\}\) is a finite collection of open sets in a topological space \((X, \mathcal{T})\), then the intersection \[ \bigcap_{i=1}^{n} U_i \] is open. #### (b) Problem Statement Provide an example of a topological space and a collection of open sets within that space to illustrate that the infinite intersection of open sets may not necessarily be open.
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