Question #8 1. Let , y ER. Determine if the following are metrics on R or not(a) d, (x, y) = V[x – y|(b) d2(x,y) = (x – y)*2. Let E CX where X is a metric space. Let E° denote all the interior points of E. (E°is an openset-you don't have to prove this, but you can use this fact to prove the following)(a) Show E is open if and only iff E = E°(b) Show: If GCE and G is open, then G C E°3. Let K,and K2 be compact subsets of a metric space X. Show K, U K2 is compactShow NaEa Ka is closed4. Let {Ka}aea be a collection of compact subsets of a metric space,in X5. (a) Consider the collection of open sets, {(-n,n)}"=1 in R with the usual metric d(x,y) =|x – y| . Use this collection of open sets to show R is not compact (make sure to prove R =Unej(-n, n) as part of your work).(b) Give an example of a collection of compact sets in R , say {Kn}n=1 such that U=1 Kn isnot compact (Hint: consider part (a) )6. Let E C X where X is a metric space. Show the limit points of E are the same as the limitpoints of Ē , the closure of E. That is, show E'= (E U E')',7. (a) Let A C X,B C X where X is a metric space. Show AnB CÃ O Ē(b) Consider A = [3,7),B = (7,9] as subset of R with the usual metric, dr (x, y) = | x – y| .Show Ā n B is not a subset of An B8. Let X be a metric space. Let A and B be separated sets in X. Show either E CA or ESBwhere E is a connected subset of A U B

Name: Question #8 1. Let , y ER. Determine if the following are metrics on R
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Question #8 1. Let , y ER. Determine if the following are metrics on R or not(a) d, (x, y) = V[x – y|(b) d2(x,y) = (x – y)*2. Let E CX where X is a metric space. Let E° denote all the interior points of E. (E°is an openset-you don't have to prove thi