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












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












    $begingroup$


    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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 6 hours ago









      Asaf Karagila♦

      306k33438769




      306k33438769










      asked 13 hours ago









      Sean ThrasherSean Thrasher

      384




      384






















          2 Answers
          2






          active

          oldest

          votes


















          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).






            share|cite|improve this answer











            $endgroup$













              Your Answer





              StackExchange.ifUsing("editor", function () {
              return StackExchange.using("mathjaxEditing", function () {
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              });
              });
              }, "mathjax-editing");

              StackExchange.ready(function() {
              var channelOptions = {
              tags: "".split(" "),
              id: "69"
              };
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function() {
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled) {
              StackExchange.using("snippets", function() {
              createEditor();
              });
              }
              else {
              createEditor();
              }
              });

              function createEditor() {
              StackExchange.prepareEditor({
              heartbeatType: 'answer',
              autoActivateHeartbeat: false,
              convertImagesToLinks: true,
              noModals: true,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              imageUploader: {
              brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
              contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
              allowUrls: true
              },
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              });


              }
              });














              draft saved

              draft discarded


















              StackExchange.ready(
              function () {
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3148005%2fprove-that-u-x-in-x-mid-dx-adx-b-is-open-when-a-and-b-are-disjoi%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              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






























                              draft saved

                              draft discarded




















































                              Thanks for contributing an answer to Mathematics Stack Exchange!


                              • Please be sure to answer the question. Provide details and share your research!

                              But avoid …



                              • Asking for help, clarification, or responding to other answers.

                              • Making statements based on opinion; back them up with references or personal experience.


                              Use MathJax to format equations. MathJax reference.


                              To learn more, see our tips on writing great answers.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function () {
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3148005%2fprove-that-u-x-in-x-mid-dx-adx-b-is-open-when-a-and-b-are-disjoi%23new-answer', 'question_page');
                              }
                              );

                              Post as a guest















                              Required, but never shown





















































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown

































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown







                              -

                              Popular posts from this blog

                              數位音樂下載

                              Image
                              數位音樂下載 為非以「實體」方式來販賣音樂的相關產品，而使用數位格式如mp3、AAC等文件格式來進行銷售販賣的一种音乐产业形式。著名的數位音樂下載服務商有iTunes Store、RecoChoku、Google Play音乐和微软Zune等。数字音乐下载在二十一世纪初一度风靡全球， [1] [2] 使得传统唱片产业走向衰落，但其自身也在2010年代流媒体音乐服务如Apple Music、Spotify等的挤压下迅速失去市场份额。 [3] [4] 数字音乐下载有便于携带、传播迅速、制作简单成本低等优点，与流媒体音乐相比也能更好保障音乐人权益； [5] 但它同时很可能带来侵犯版权和滥用的情形， [6] 也给音乐产业带来了诸多不利影响。数字音乐下载为音乐人和乐迷提供了更加便捷直接的交流渠道，也一定程度上推动了网络音乐人的出现和独立音乐的迅速发展。 [7] 目录 1 历史 1.1 技术积淀（1987-2000） 1.2 出现与兴起（2000-2004） 1.3 唱片市场的颠覆者（2004-2007） 1.4 巅峰和衰落（2005-2013） 1.5 摇摇欲坠的现状与不可预知的明天（2014至今） 2 主要文件格式 2.1 PCM编码格式 2.1.1 WAV 2.1.2 APE 2.1.3 FLAC 2.2 苹果推出的无损文件格式 2.2.1 AIFF 2.2.2 ALAC 2.3 有损压缩格式 2.3.1 MP3 2.3.2 AAC 3 主要市场 3.1 iTunes Store 3.2 Google Play音乐商店 3.3 亚马逊音乐商店 3.4 部分其他音乐商店 4 影响与评价 4.1 对音乐产业的影响 4.2 对科技行业的影响 4.3 正面评价 4.4 负面评价 4.4.1 侵权问题 4.4.2 经营模式不明晰 4.4.3 不适应移动时代 4.5 文化现象 5 延伸阅读 6 参考资料 历史 苹果iTunes+iPod的软硬件搭配让数字音...
                              Read more

                              When can things happen in Etherscan, such as the picture below?

                              Image
                              2 I don't know why balance of the address is 0. when can this situation? etherscan balances share | improve this question asked yesterday Ru Hi Ru Hi 11 1 New contributor Ru Hi is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct. add a comment  |  ...
                              Read more

                              格利澤436b

                              Image
                              格利泽436b 太陽系外行星 太陽系外行星列表 艺术家笔下的格利泽436b 母恆星 母恆星 Gliese 436 星座 獅子座 赤經 ( α ) 11 h  42 m  11.0941 s [1] 赤緯 ( δ ) +26° 42′ 23.652″ [1] 距離 33.4 ly (10.2 pc) 光譜類型 M2.5 V [1] 軌道參數 半长轴 ( a ) 0.0291±0.0004 [2] AU 軌道離心率 ( e ) 0.150±0.012 [2] 公轉週期 ( P ) 2.643904±0.000005 [3] d (0.00723849 y) 軌道傾角 ( i ) 85.8 +0.21 −0.25 [3] ° 角距 ( θ ) 2.794 mas 近星點時間 ( T 0 ) 2,451,551.716 ±0.01 JD 半振幅 ( K ) 18.68±0.8 m/s 物理性质 质量 ( m ) 22.2±1.0 [2] M ⊕ 半徑 ( r ) 4.327±0.183 [2] [4] R ⊕ 密度 ( ρ ) 1.51 g cm -3 表面重力 ( g ) 1.18 g 溫度 ( T ) 712±36 [2] K 發現 發現時間 2004 發現者 巴特勒、沃格特、 馬西 et al. 發現方法 徑向速度、凌日法 發現地點 美國加利福尼亞州 發表論文 出版 其他名稱 Ross 905 b, GJ 436 b, [5] LTT 13213 b, GCTP 2704.10 b, LHS 310, AC+27:28217 b, Vyssotsky 616 b, HIP 57087 b, GEN# +9.80120068 b, LP 319-75 b, G 121-7 b, LSPM J1142+2642 b, 1RXS J114211.9+264328 b, ASCC 683818 b, G 147-68 b, UCAC2 41198281 b, BP...
                              Read more