Prove that $U={x in Xmid d(x,A)<d(x,B)}$ is open when $A$ and $B$ are disjoint












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$begingroup$


In a metric space $(X,d)$, I have the set $$U={x in Xmid d(x,A)<d(x,B)}$$
where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C={xin Xmid d(x,A)=d(x,C)}$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!






metric-spaces







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$endgroup$

















    4












    $begingroup$


    In a metric space $(X,d)$, I have the set $$U={x in Xmid d(x,A)<d(x,B)}$$
    where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C={xin Xmid d(x,A)=d(x,C)}$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!






    metric-spaces







    share|cite|improve this question











    $endgroup$















      4












      4








      4





      $begingroup$


      In a metric space $(X,d)$, I have the set $$U={x in Xmid d(x,A)<d(x,B)}$$
      where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C={xin Xmid d(x,A)=d(x,C)}$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!






      metric-spaces







      share|cite|improve this question











      $endgroup$




      In a metric space $(X,d)$, I have the set $$U={x in Xmid d(x,A)<d(x,B)}$$
      where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C={xin Xmid d(x,A)=d(x,C)}$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!






      metric-spaces




      metric-spaces






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      share|cite|improve this question













      share|cite|improve this question




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      edited 6 hours ago









      Asaf Karagila♦

      306k33438769




      306k33438769










      asked 13 hours ago









      Sean ThrasherSean Thrasher

      384




      384






















          2 Answers
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          10












          $begingroup$

          $f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$






          share|cite|improve this answer









          $endgroup$





















            5












            $begingroup$

            Your strategy will not work in an arbitrary metric space. Consider for example
            $ X = mathbb R setminus{0}$ with the usual distance and then $A={-1}$, $B={1}$. Then your $C$ is empty.



            Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.



            (Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).






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              2 Answers
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              2 Answers
              2






              active

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              active

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              active

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              10












              $begingroup$

              $f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$






              share|cite|improve this answer









              $endgroup$


















                10












                $begingroup$

                $f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$






                share|cite|improve this answer









                $endgroup$
















                  10












                  10








                  10





                  $begingroup$

                  $f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$






                  share|cite|improve this answer









                  $endgroup$



                  $f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 12 hours ago









                  Tsemo AristideTsemo Aristide

                  59.4k11446




                  59.4k11446























                      5












                      $begingroup$

                      Your strategy will not work in an arbitrary metric space. Consider for example
                      $ X = mathbb R setminus{0}$ with the usual distance and then $A={-1}$, $B={1}$. Then your $C$ is empty.



                      Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.



                      (Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).






                      share|cite|improve this answer











                      $endgroup$


















                        5












                        $begingroup$

                        Your strategy will not work in an arbitrary metric space. Consider for example
                        $ X = mathbb R setminus{0}$ with the usual distance and then $A={-1}$, $B={1}$. Then your $C$ is empty.



                        Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.



                        (Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).






                        share|cite|improve this answer











                        $endgroup$
















                          5












                          5








                          5





                          $begingroup$

                          Your strategy will not work in an arbitrary metric space. Consider for example
                          $ X = mathbb R setminus{0}$ with the usual distance and then $A={-1}$, $B={1}$. Then your $C$ is empty.



                          Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.



                          (Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).






                          share|cite|improve this answer











                          $endgroup$



                          Your strategy will not work in an arbitrary metric space. Consider for example
                          $ X = mathbb R setminus{0}$ with the usual distance and then $A={-1}$, $B={1}$. Then your $C$ is empty.



                          Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.



                          (Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited 12 hours ago

























                          answered 12 hours ago









                          Henning MakholmHenning Makholm

                          242k17308549




                          242k17308549






























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