Mixing
Mathematics- Topology [closed]
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Wt is totally bounded set?
[0,1] is it totally bounded set?
As per definition X is equal to collection of all all open spheres with centers are the points of X
Then how can we write [0,1] in that way..
One more, X is compact iff X is complete and totally bounded
From this statement i got confusion about totally bounded sets definition
real-analysis
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
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closed as unclear what you're asking by Andrés E. Caicedo, zipirovich, verret, Cesareo, max_zorn Jan 21 at 3:57
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
Wt is totally bounded set?
[0,1] is it totally bounded set?
As per definition X is equal to collection of all all open spheres with centers are the points of X
Then how can we write [0,1] in that way..
One more, X is compact iff X is complete and totally bounded
From this statement i got confusion about totally bounded sets definition
real-analysis
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
$endgroup$
closed as unclear what you're asking by Andrés E. Caicedo, zipirovich, verret, Cesareo, max_zorn Jan 21 at 3:57
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
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Welcome to M.SE! This is a question and answer site for people studying math at any level and professionals in related fields. We are here to help us each others. However, you should avoid questions that can be answered with a simple search on the web. For example here.
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– Dog_69
Jan 20 at 17:35
add a comment |
$begingroup$
Wt is totally bounded set?
[0,1] is it totally bounded set?
As per definition X is equal to collection of all all open spheres with centers are the points of X
Then how can we write [0,1] in that way..
One more, X is compact iff X is complete and totally bounded
From this statement i got confusion about totally bounded sets definition
real-analysis
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
$endgroup$
Wt is totally bounded set?
[0,1] is it totally bounded set?
As per definition X is equal to collection of all all open spheres with centers are the points of X
Then how can we write [0,1] in that way..
One more, X is compact iff X is complete and totally bounded
From this statement i got confusion about totally bounded sets definition
real-analysis
real-analysis
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
asked Jan 20 at 17:20
routhu vinodhrouthu vinodh
6
6
closed as unclear what you're asking by Andrés E. Caicedo, zipirovich, verret, Cesareo, max_zorn Jan 21 at 3:57
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Andrés E. Caicedo, zipirovich, verret, Cesareo, max_zorn Jan 21 at 3:57
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
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Welcome to M.SE! This is a question and answer site for people studying math at any level and professionals in related fields. We are here to help us each others. However, you should avoid questions that can be answered with a simple search on the web. For example here.
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– Dog_69
Jan 20 at 17:35
add a comment |
3
$begingroup$
Welcome to M.SE! This is a question and answer site for people studying math at any level and professionals in related fields. We are here to help us each others. However, you should avoid questions that can be answered with a simple search on the web. For example here.
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– Dog_69
Jan 20 at 17:35
3
3
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Welcome to M.SE! This is a question and answer site for people studying math at any level and professionals in related fields. We are here to help us each others. However, you should avoid questions that can be answered with a simple search on the web. For example here.
$endgroup$
– Dog_69
Jan 20 at 17:35
$begingroup$
Welcome to M.SE! This is a question and answer site for people studying math at any level and professionals in related fields. We are here to help us each others. However, you should avoid questions that can be answered with a simple search on the web. For example here.
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– Dog_69
Jan 20 at 17:35
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1 Answer
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A set $A$ is called totally bounded set if for every $epsilon >0 $ there is a finite collection of open balls $mathcal{B} = {B_{1}, cdots , B_{n}} $ with radii at most $ epsilon$ such that $$ A subset bigcup_{k=1}^{n} {B_{k}} $$
For the case of the set $[0,1] $ , for given $epsilon > 0 $, take $$mathcal{B} ={ B_{k} = ((k-1)epsilon, (k+1)epsilon) , : k= 0,cdots ,ceil(frac{1}{epsilon})} $$
Then it is clear that $$ bigcup_{k=0}^{n} {B_{k}} = (-epsilon, 1+ epsilon) supset [0, 1] $$ So $[0, 1] $ is totally bounded set in $mathbb{R} $ Since $ |mathcal{B}|< infty $.
For the second question. It is the major implication of compactness property.
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A set $A$ is called totally bounded set if for every $epsilon >0 $ there is a finite collection of open balls $mathcal{B} = {B_{1}, cdots , B_{n}} $ with radii at most $ epsilon$ such that $$ A subset bigcup_{k=1}^{n} {B_{k}} $$
For the case of the set $[0,1] $ , for given $epsilon > 0 $, take $$mathcal{B} ={ B_{k} = ((k-1)epsilon, (k+1)epsilon) , : k= 0,cdots ,ceil(frac{1}{epsilon})} $$
Then it is clear that $$ bigcup_{k=0}^{n} {B_{k}} = (-epsilon, 1+ epsilon) supset [0, 1] $$ So $[0, 1] $ is totally bounded set in $mathbb{R} $ Since $ |mathcal{B}|< infty $.
For the second question. It is the major implication of compactness property.
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
$endgroup$
add a comment |
$begingroup$
A set $A$ is called totally bounded set if for every $epsilon >0 $ there is a finite collection of open balls $mathcal{B} = {B_{1}, cdots , B_{n}} $ with radii at most $ epsilon$ such that $$ A subset bigcup_{k=1}^{n} {B_{k}} $$
For the case of the set $[0,1] $ , for given $epsilon > 0 $, take $$mathcal{B} ={ B_{k} = ((k-1)epsilon, (k+1)epsilon) , : k= 0,cdots ,ceil(frac{1}{epsilon})} $$
Then it is clear that $$ bigcup_{k=0}^{n} {B_{k}} = (-epsilon, 1+ epsilon) supset [0, 1] $$ So $[0, 1] $ is totally bounded set in $mathbb{R} $ Since $ |mathcal{B}|< infty $.
For the second question. It is the major implication of compactness property.
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
$endgroup$
add a comment |
$begingroup$
A set $A$ is called totally bounded set if for every $epsilon >0 $ there is a finite collection of open balls $mathcal{B} = {B_{1}, cdots , B_{n}} $ with radii at most $ epsilon$ such that $$ A subset bigcup_{k=1}^{n} {B_{k}} $$
For the case of the set $[0,1] $ , for given $epsilon > 0 $, take $$mathcal{B} ={ B_{k} = ((k-1)epsilon, (k+1)epsilon) , : k= 0,cdots ,ceil(frac{1}{epsilon})} $$
Then it is clear that $$ bigcup_{k=0}^{n} {B_{k}} = (-epsilon, 1+ epsilon) supset [0, 1] $$ So $[0, 1] $ is totally bounded set in $mathbb{R} $ Since $ |mathcal{B}|< infty $.
For the second question. It is the major implication of compactness property.
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
$endgroup$
A set $A$ is called totally bounded set if for every $epsilon >0 $ there is a finite collection of open balls $mathcal{B} = {B_{1}, cdots , B_{n}} $ with radii at most $ epsilon$ such that $$ A subset bigcup_{k=1}^{n} {B_{k}} $$
For the case of the set $[0,1] $ , for given $epsilon > 0 $, take $$mathcal{B} ={ B_{k} = ((k-1)epsilon, (k+1)epsilon) , : k= 0,cdots ,ceil(frac{1}{epsilon})} $$
Then it is clear that $$ bigcup_{k=0}^{n} {B_{k}} = (-epsilon, 1+ epsilon) supset [0, 1] $$ So $[0, 1] $ is totally bounded set in $mathbb{R} $ Since $ |mathcal{B}|< infty $.
For the second question. It is the major implication of compactness property.
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
edited Jan 20 at 19:26
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
answered Jan 20 at 17:58
abdullah azzamabdullah azzam
294
294
add a comment |
add a comment |
3
$begingroup$
Welcome to M.SE! This is a question and answer site for people studying math at any level and professionals in related fields. We are here to help us each others. However, you should avoid questions that can be answered with a simple search on the web. For example here.
$endgroup$
– Dog_69
Jan 20 at 17:35