Compactness and Enlargement
$begingroup$
Consider a bounded continuous function that satisfies $|f(x)|leqepsilon$ on a compact set K. I am asked to prove that there is a $delta$ enlargement of K, $K^delta$ such that
$$|f(x)|leq2epsilon ,forall xin K^delta$$
$K^delta={x|d(x,K)<delta}$ for some metric.
The hint says uses the compactness of K. I was trying to use the finite covering property of K but the argument does not go through when I try to select a $delta$. I would like to know if there is some alternative way to prove the above claim.
general-topology analysis
asked Dec 27 '18 at 16:18
displayname2displayname2
505
$endgroup$
add a comment |
$begingroup$
Consider a bounded continuous function that satisfies $|f(x)|leqepsilon$ on a compact set K. I am asked to prove that there is a $delta$ enlargement of K, $K^delta$ such that
$$|f(x)|leq2epsilon ,forall xin K^delta$$
$K^delta={x|d(x,K)<delta}$ for some metric.
The hint says uses the compactness of K. I was trying to use the finite covering property of K but the argument does not go through when I try to select a $delta$. I would like to know if there is some alternative way to prove the above claim.
general-topology analysis
asked Dec 27 '18 at 16:18
displayname2displayname2
505
$endgroup$
$begingroup$
What is the domain and range of $f$?
$endgroup$
– freakish
Dec 27 '18 at 16:29
$begingroup$
there is no specification of that. x is in a general metric space D and f is a bounded and continuous function.
$endgroup$
– displayname2
Dec 27 '18 at 16:35
add a comment |
$begingroup$
Consider a bounded continuous function that satisfies $|f(x)|leqepsilon$ on a compact set K. I am asked to prove that there is a $delta$ enlargement of K, $K^delta$ such that
$$|f(x)|leq2epsilon ,forall xin K^delta$$
$K^delta={x|d(x,K)<delta}$ for some metric.
The hint says uses the compactness of K. I was trying to use the finite covering property of K but the argument does not go through when I try to select a $delta$. I would like to know if there is some alternative way to prove the above claim.
general-topology analysis
asked Dec 27 '18 at 16:18
displayname2displayname2
505
$endgroup$
Consider a bounded continuous function that satisfies $|f(x)|leqepsilon$ on a compact set K. I am asked to prove that there is a $delta$ enlargement of K, $K^delta$ such that
$$|f(x)|leq2epsilon ,forall xin K^delta$$
$K^delta={x|d(x,K)<delta}$ for some metric.
The hint says uses the compactness of K. I was trying to use the finite covering property of K but the argument does not go through when I try to select a $delta$. I would like to know if there is some alternative way to prove the above claim.
general-topology analysis
general-topology analysis
asked Dec 27 '18 at 16:18
displayname2displayname2
505
asked Dec 27 '18 at 16:18
displayname2displayname2
505
asked Dec 27 '18 at 16:18
displayname2displayname2
505
asked Dec 27 '18 at 16:18
displayname2displayname2
505
asked Dec 27 '18 at 16:18
displayname2displayname2
505
505
$begingroup$
What is the domain and range of $f$?
$endgroup$
– freakish
Dec 27 '18 at 16:29
$begingroup$
there is no specification of that. x is in a general metric space D and f is a bounded and continuous function.
$endgroup$
– displayname2
Dec 27 '18 at 16:35
add a comment |
$begingroup$
What is the domain and range of $f$?
$endgroup$
– freakish
Dec 27 '18 at 16:29
$begingroup$
there is no specification of that. x is in a general metric space D and f is a bounded and continuous function.
$endgroup$
– displayname2
Dec 27 '18 at 16:35
$begingroup$
What is the domain and range of $f$?
$endgroup$
– freakish
Dec 27 '18 at 16:29
$begingroup$
What is the domain and range of $f$?
$endgroup$
– freakish
Dec 27 '18 at 16:29
$begingroup$
there is no specification of that. x is in a general metric space D and f is a bounded and continuous function.
$endgroup$
– displayname2
Dec 27 '18 at 16:35
$begingroup$
there is no specification of that. x is in a general metric space D and f is a bounded and continuous function.
$endgroup$
– displayname2
Dec 27 '18 at 16:35
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: for each $xin K$ there is a $delta_x>0$ such that $yin B(x,delta_x)Rightarrow f(y)in B(f(x),epsilon).$ Then, $Ksubseteq bigcup_{xin K}B(x,delta_x)$ and $mathscr A={B(x,delta_x):xin K}$ is a cover of $K$. Reduce to a finite subcover and unravel the definitions.
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
$endgroup$
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
1
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: for each $xin K$ there is a $delta_x>0$ such that $yin B(x,delta_x)Rightarrow f(y)in B(f(x),epsilon).$ Then, $Ksubseteq bigcup_{xin K}B(x,delta_x)$ and $mathscr A={B(x,delta_x):xin K}$ is a cover of $K$. Reduce to a finite subcover and unravel the definitions.
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
$endgroup$
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
1
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
add a comment |
$begingroup$
Hint: for each $xin K$ there is a $delta_x>0$ such that $yin B(x,delta_x)Rightarrow f(y)in B(f(x),epsilon).$ Then, $Ksubseteq bigcup_{xin K}B(x,delta_x)$ and $mathscr A={B(x,delta_x):xin K}$ is a cover of $K$. Reduce to a finite subcover and unravel the definitions.
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
$endgroup$
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
1
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
add a comment |
$begingroup$
Hint: for each $xin K$ there is a $delta_x>0$ such that $yin B(x,delta_x)Rightarrow f(y)in B(f(x),epsilon).$ Then, $Ksubseteq bigcup_{xin K}B(x,delta_x)$ and $mathscr A={B(x,delta_x):xin K}$ is a cover of $K$. Reduce to a finite subcover and unravel the definitions.
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
$endgroup$
Hint: for each $xin K$ there is a $delta_x>0$ such that $yin B(x,delta_x)Rightarrow f(y)in B(f(x),epsilon).$ Then, $Ksubseteq bigcup_{xin K}B(x,delta_x)$ and $mathscr A={B(x,delta_x):xin K}$ is a cover of $K$. Reduce to a finite subcover and unravel the definitions.
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
answered Dec 27 '18 at 16:29
MatematletaMatematleta
11.5k2920
11.5k2920
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
1
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
add a comment |
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
1
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
Then my question is that suppose we have picked the finite cover and found the minimum radius $delta_{x^*}$ for example. How can we make sure that K^{delta_{x^*}} is what we want? There may be points that are included in the finite subcover but their $delta_{x^*}$ balls does not satisfy the desired criterion.
$endgroup$
– displayname2
Dec 27 '18 at 16:32
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
$begingroup$
because if $xin K$ then $yin B(x,frac{1}{2}delta_x)Rightarrow d(y,K)le d(y,x)<delta_xRightarrow d(f(y),f(x))<2epsilon $.
$endgroup$
– Matematleta
Dec 27 '18 at 16:38
1
1
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are right! I think that's the step I am missing. Thank you!
$endgroup$
– displayname2
Dec 27 '18 at 16:41
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
$begingroup$
you are welcome!
$endgroup$
– Matematleta
Dec 27 '18 at 16:48
add a comment |
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$begingroup$
What is the domain and range of $f$?
$endgroup$
– freakish
Dec 27 '18 at 16:29
$begingroup$
there is no specification of that. x is in a general metric space D and f is a bounded and continuous function.
$endgroup$
– displayname2
Dec 27 '18 at 16:35