Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![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 open
set-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 compact
Show NaEa Ka is closed
4. Let {Ka}aea be a collection of compact subsets of a metric space,
in X
5. (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 is
not 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 limit
points 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 B
8. Let X be a metric space. Let A and B be separated sets in X. Show either E CA or ESB
where E is a connected subset of A U B](https://content.bartleby.com/qna-images/question/cfc5bdab-6654-444e-b9d5-5fc4b8470fbb/9007f168-56ae-436b-9289-cf0e50c98243/g65hk5g_processed.jpeg)
Transcribed Image Text: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 open
set-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 compact
Show NaEa Ka is closed
4. Let {Ka}aea be a collection of compact subsets of a metric space,
in X
5. (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 is
not 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 limit
points 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 B
8. Let X be a metric space. Let A and B be separated sets in X. Show either E CA or ESB
where E is a connected subset of A U B
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