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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 (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)
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.
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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