$A$ is compact and closed then prove …












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I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two points in $A$ and another arbitrary set?
$A$ is compact and closed then prove there exist an $a$ member of $A$ and $b$ a member of $B$ such that $d(a,b) = d(x,y)$ ($x$ is a member of $A$ and $y$ a member of $B$).










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    -1












    $begingroup$


    I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two points in $A$ and another arbitrary set?
    $A$ is compact and closed then prove there exist an $a$ member of $A$ and $b$ a member of $B$ such that $d(a,b) = d(x,y)$ ($x$ is a member of $A$ and $y$ a member of $B$).










    share|cite|improve this question











    $endgroup$















      -1












      -1








      -1





      $begingroup$


      I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two points in $A$ and another arbitrary set?
      $A$ is compact and closed then prove there exist an $a$ member of $A$ and $b$ a member of $B$ such that $d(a,b) = d(x,y)$ ($x$ is a member of $A$ and $y$ a member of $B$).










      share|cite|improve this question











      $endgroup$




      I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two points in $A$ and another arbitrary set?
      $A$ is compact and closed then prove there exist an $a$ member of $A$ and $b$ a member of $B$ such that $d(a,b) = d(x,y)$ ($x$ is a member of $A$ and $y$ a member of $B$).







      real-analysis compactness closed-form






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      edited Dec 10 '18 at 6:32









      Ali

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      asked Dec 10 '18 at 6:04









      PegiPegi

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          $begingroup$

          For $A,B$ two non-empty sets, $d(A,B):= inf {d(x,y): x in A, y in B}$ always exists, as we have a set of real numbers that is non-empty and bounded below (by $0$ trivially). The same holds for $d(x,B) = inf {d(x,b); b in B}$ as a special case.



          However, we can note that the function $f_B: x to d(x,B)$ is continuous for $x in X$ (as $|d(x,B) - d(x', B)| le d(x,x')$ by the triangle inequality, so the function is contractive hence uniformly continuous) and so $f_B$ assumes a minimum and maximum value on the compact subset $A$.



          If we do not assume anything on $B$ we need not always have such $a,b$ as asked for: $A = [0,1], B = (1,2)$ is an example where $d(A,B)=0$ and $A$ is compact. If $B$ too is compact we do have such $a,b$, e.g. In many cases $B$ being closed is also sufficient besides $A$ compact.






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            $begingroup$

            For $A,B$ two non-empty sets, $d(A,B):= inf {d(x,y): x in A, y in B}$ always exists, as we have a set of real numbers that is non-empty and bounded below (by $0$ trivially). The same holds for $d(x,B) = inf {d(x,b); b in B}$ as a special case.



            However, we can note that the function $f_B: x to d(x,B)$ is continuous for $x in X$ (as $|d(x,B) - d(x', B)| le d(x,x')$ by the triangle inequality, so the function is contractive hence uniformly continuous) and so $f_B$ assumes a minimum and maximum value on the compact subset $A$.



            If we do not assume anything on $B$ we need not always have such $a,b$ as asked for: $A = [0,1], B = (1,2)$ is an example where $d(A,B)=0$ and $A$ is compact. If $B$ too is compact we do have such $a,b$, e.g. In many cases $B$ being closed is also sufficient besides $A$ compact.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              For $A,B$ two non-empty sets, $d(A,B):= inf {d(x,y): x in A, y in B}$ always exists, as we have a set of real numbers that is non-empty and bounded below (by $0$ trivially). The same holds for $d(x,B) = inf {d(x,b); b in B}$ as a special case.



              However, we can note that the function $f_B: x to d(x,B)$ is continuous for $x in X$ (as $|d(x,B) - d(x', B)| le d(x,x')$ by the triangle inequality, so the function is contractive hence uniformly continuous) and so $f_B$ assumes a minimum and maximum value on the compact subset $A$.



              If we do not assume anything on $B$ we need not always have such $a,b$ as asked for: $A = [0,1], B = (1,2)$ is an example where $d(A,B)=0$ and $A$ is compact. If $B$ too is compact we do have such $a,b$, e.g. In many cases $B$ being closed is also sufficient besides $A$ compact.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                For $A,B$ two non-empty sets, $d(A,B):= inf {d(x,y): x in A, y in B}$ always exists, as we have a set of real numbers that is non-empty and bounded below (by $0$ trivially). The same holds for $d(x,B) = inf {d(x,b); b in B}$ as a special case.



                However, we can note that the function $f_B: x to d(x,B)$ is continuous for $x in X$ (as $|d(x,B) - d(x', B)| le d(x,x')$ by the triangle inequality, so the function is contractive hence uniformly continuous) and so $f_B$ assumes a minimum and maximum value on the compact subset $A$.



                If we do not assume anything on $B$ we need not always have such $a,b$ as asked for: $A = [0,1], B = (1,2)$ is an example where $d(A,B)=0$ and $A$ is compact. If $B$ too is compact we do have such $a,b$, e.g. In many cases $B$ being closed is also sufficient besides $A$ compact.






                share|cite|improve this answer









                $endgroup$



                For $A,B$ two non-empty sets, $d(A,B):= inf {d(x,y): x in A, y in B}$ always exists, as we have a set of real numbers that is non-empty and bounded below (by $0$ trivially). The same holds for $d(x,B) = inf {d(x,b); b in B}$ as a special case.



                However, we can note that the function $f_B: x to d(x,B)$ is continuous for $x in X$ (as $|d(x,B) - d(x', B)| le d(x,x')$ by the triangle inequality, so the function is contractive hence uniformly continuous) and so $f_B$ assumes a minimum and maximum value on the compact subset $A$.



                If we do not assume anything on $B$ we need not always have such $a,b$ as asked for: $A = [0,1], B = (1,2)$ is an example where $d(A,B)=0$ and $A$ is compact. If $B$ too is compact we do have such $a,b$, e.g. In many cases $B$ being closed is also sufficient besides $A$ compact.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 10 '18 at 6:47









                Henno BrandsmaHenno Brandsma

                108k347114




                108k347114






























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