Why is a polar cone a closed set?












5












$begingroup$


Let $X subset mathbb{R}^n$. We define the polar cone as



$$Xº:={xinmathbb{R}^n,|,langle u,xrangleleq 0,forall uin X}$$



How can I show that this set is closed?



If I fix some $uin X$ then I have that ${xinmathbb{R}^n,|,langle u,xrangleleq 0}$ is a closed halfspace; but if $X$ is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).






general-topology convex-analysis







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    What does $u'x$ mean?
    $endgroup$
    – José Carlos Santos
    15 hours ago












  • $begingroup$
    probably inter product with $u'$ the tranpose
    $endgroup$
    – dmtri
    15 hours ago












  • $begingroup$
    @JoséCarlosSantos Usual product in $mathbb{R^n}$. Edited.
    $endgroup$
    – Lecter
    15 hours ago






  • 1




    $begingroup$
    why the intersection of closed sets is not a close set?
    $endgroup$
    – dmtri
    15 hours ago






  • 1




    $begingroup$
    @dmtri It's done.
    $endgroup$
    – José Carlos Santos
    15 hours ago
















5












$begingroup$


Let $X subset mathbb{R}^n$. We define the polar cone as



$$Xº:={xinmathbb{R}^n,|,langle u,xrangleleq 0,forall uin X}$$



How can I show that this set is closed?



If I fix some $uin X$ then I have that ${xinmathbb{R}^n,|,langle u,xrangleleq 0}$ is a closed halfspace; but if $X$ is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).






general-topology convex-analysis







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    What does $u'x$ mean?
    $endgroup$
    – José Carlos Santos
    15 hours ago












  • $begingroup$
    probably inter product with $u'$ the tranpose
    $endgroup$
    – dmtri
    15 hours ago












  • $begingroup$
    @JoséCarlosSantos Usual product in $mathbb{R^n}$. Edited.
    $endgroup$
    – Lecter
    15 hours ago






  • 1




    $begingroup$
    why the intersection of closed sets is not a close set?
    $endgroup$
    – dmtri
    15 hours ago






  • 1




    $begingroup$
    @dmtri It's done.
    $endgroup$
    – José Carlos Santos
    15 hours ago














5












5








5


1



$begingroup$


Let $X subset mathbb{R}^n$. We define the polar cone as



$$Xº:={xinmathbb{R}^n,|,langle u,xrangleleq 0,forall uin X}$$



How can I show that this set is closed?



If I fix some $uin X$ then I have that ${xinmathbb{R}^n,|,langle u,xrangleleq 0}$ is a closed halfspace; but if $X$ is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).






general-topology convex-analysis







share|cite|improve this question











$endgroup$




Let $X subset mathbb{R}^n$. We define the polar cone as



$$Xº:={xinmathbb{R}^n,|,langle u,xrangleleq 0,forall uin X}$$



How can I show that this set is closed?



If I fix some $uin X$ then I have that ${xinmathbb{R}^n,|,langle u,xrangleleq 0}$ is a closed halfspace; but if $X$ is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).






general-topology convex-analysis




general-topology convex-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 15 hours ago









Rodrigo de Azevedo

13.1k41960




13.1k41960










asked 15 hours ago









LecterLecter

11210




11210








  • 2




    $begingroup$
    What does $u'x$ mean?
    $endgroup$
    – José Carlos Santos
    15 hours ago












  • $begingroup$
    probably inter product with $u'$ the tranpose
    $endgroup$
    – dmtri
    15 hours ago












  • $begingroup$
    @JoséCarlosSantos Usual product in $mathbb{R^n}$. Edited.
    $endgroup$
    – Lecter
    15 hours ago






  • 1




    $begingroup$
    why the intersection of closed sets is not a close set?
    $endgroup$
    – dmtri
    15 hours ago






  • 1




    $begingroup$
    @dmtri It's done.
    $endgroup$
    – José Carlos Santos
    15 hours ago














  • 2




    $begingroup$
    What does $u'x$ mean?
    $endgroup$
    – José Carlos Santos
    15 hours ago












  • $begingroup$
    probably inter product with $u'$ the tranpose
    $endgroup$
    – dmtri
    15 hours ago












  • $begingroup$
    @JoséCarlosSantos Usual product in $mathbb{R^n}$. Edited.
    $endgroup$
    – Lecter
    15 hours ago






  • 1




    $begingroup$
    why the intersection of closed sets is not a close set?
    $endgroup$
    – dmtri
    15 hours ago






  • 1




    $begingroup$
    @dmtri It's done.
    $endgroup$
    – José Carlos Santos
    15 hours ago








2




2




$begingroup$
What does $u'x$ mean?
$endgroup$
– José Carlos Santos
15 hours ago






$begingroup$
What does $u'x$ mean?
$endgroup$
– José Carlos Santos
15 hours ago














$begingroup$
probably inter product with $u'$ the tranpose
$endgroup$
– dmtri
15 hours ago






$begingroup$
probably inter product with $u'$ the tranpose
$endgroup$
– dmtri
15 hours ago














$begingroup$
@JoséCarlosSantos Usual product in $mathbb{R^n}$. Edited.
$endgroup$
– Lecter
15 hours ago




$begingroup$
@JoséCarlosSantos Usual product in $mathbb{R^n}$. Edited.
$endgroup$
– Lecter
15 hours ago




1




1




$begingroup$
why the intersection of closed sets is not a close set?
$endgroup$
– dmtri
15 hours ago




$begingroup$
why the intersection of closed sets is not a close set?
$endgroup$
– dmtri
15 hours ago




1




1




$begingroup$
@dmtri It's done.
$endgroup$
– José Carlos Santos
15 hours ago




$begingroup$
@dmtri It's done.
$endgroup$
– José Carlos Santos
15 hours ago










2 Answers
2






active

oldest

votes


















7












$begingroup$

Actually, your argument works: $X^0$ is closed because it can be expressed as an intersection of closed sets.






share|cite|improve this answer









$endgroup$





















    7












    $begingroup$


    if 𝑋 is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).




    Yes you can. One of the axioms of topology is that any union of open sets is open. (This is easy to show directly for the standard topology on a metric space).



    Taking complements, you get that any intersection of closed sets is closed.






    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%2f3151318%2fwhy-is-a-polar-cone-a-closed-set%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









      7












      $begingroup$

      Actually, your argument works: $X^0$ is closed because it can be expressed as an intersection of closed sets.






      share|cite|improve this answer









      $endgroup$


















        7












        $begingroup$

        Actually, your argument works: $X^0$ is closed because it can be expressed as an intersection of closed sets.






        share|cite|improve this answer









        $endgroup$
















          7












          7








          7





          $begingroup$

          Actually, your argument works: $X^0$ is closed because it can be expressed as an intersection of closed sets.






          share|cite|improve this answer









          $endgroup$



          Actually, your argument works: $X^0$ is closed because it can be expressed as an intersection of closed sets.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 15 hours ago









          José Carlos SantosJosé Carlos Santos

          168k22132236




          168k22132236























              7












              $begingroup$


              if 𝑋 is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).




              Yes you can. One of the axioms of topology is that any union of open sets is open. (This is easy to show directly for the standard topology on a metric space).



              Taking complements, you get that any intersection of closed sets is closed.






              share|cite|improve this answer









              $endgroup$


















                7












                $begingroup$


                if 𝑋 is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).




                Yes you can. One of the axioms of topology is that any union of open sets is open. (This is easy to show directly for the standard topology on a metric space).



                Taking complements, you get that any intersection of closed sets is closed.






                share|cite|improve this answer









                $endgroup$
















                  7












                  7








                  7





                  $begingroup$


                  if 𝑋 is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).




                  Yes you can. One of the axioms of topology is that any union of open sets is open. (This is easy to show directly for the standard topology on a metric space).



                  Taking complements, you get that any intersection of closed sets is closed.






                  share|cite|improve this answer









                  $endgroup$




                  if 𝑋 is infinite we can't conclude that the intersection of closed sets is also a closed set (as far as we are talking in terms of usual topology).




                  Yes you can. One of the axioms of topology is that any union of open sets is open. (This is easy to show directly for the standard topology on a metric space).



                  Taking complements, you get that any intersection of closed sets is closed.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 15 hours ago









                  Henning MakholmHenning Makholm

                  242k17308550




                  242k17308550






























                      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%2f3151318%2fwhy-is-a-polar-cone-a-closed-set%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