Compactness and Enlargement












0












$begingroup$


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







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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
















0












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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


















  • $begingroup$
    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










1 Answer
1






active

oldest

votes


















2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















2












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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














2












2








2





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 27 '18 at 16:29









MatematletaMatematleta

12.2k21020




12.2k21020












  • $begingroup$
    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












  • $begingroup$
    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


















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