If a collection of closed sets of arbitrary cardinality in a metric space has empty intersection, does some...












4












$begingroup$


In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



Is this possible in a metric space?






general-topology metric-spaces







share|cite|improve this question









$endgroup$

















    4












    $begingroup$


    In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



    Is this possible in a metric space?






    general-topology metric-spaces







    share|cite|improve this question









    $endgroup$















      4












      4








      4


      2



      $begingroup$


      In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



      Is this possible in a metric space?






      general-topology metric-spaces







      share|cite|improve this question









      $endgroup$




      In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



      Is this possible in a metric space?






      general-topology metric-spaces




      general-topology metric-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      Matt SamuelMatt Samuel

      38.3k63768




      38.3k63768






















          2 Answers
          2






          active

          oldest

          votes


















          5












          $begingroup$

          Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Of course. Thank you.
            $endgroup$
            – Matt Samuel
            3 hours ago










          • $begingroup$
            I'm glad I could help.
            $endgroup$
            – José Carlos Santos
            3 hours ago



















          3












          $begingroup$

          The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



          Hence Santos' example was the standard example of a non-separable metric space.






          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%2f3106782%2fif-a-collection-of-closed-sets-of-arbitrary-cardinality-in-a-metric-space-has-em%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









            5












            $begingroup$

            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              3 hours ago










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              3 hours ago
















            5












            $begingroup$

            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              3 hours ago










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              3 hours ago














            5












            5








            5





            $begingroup$

            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






            share|cite|improve this answer









            $endgroup$



            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 4 hours ago









            José Carlos SantosJosé Carlos Santos

            160k22127232




            160k22127232












            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              3 hours ago










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              3 hours ago


















            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              3 hours ago










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              3 hours ago
















            $begingroup$
            Of course. Thank you.
            $endgroup$
            – Matt Samuel
            3 hours ago




            $begingroup$
            Of course. Thank you.
            $endgroup$
            – Matt Samuel
            3 hours ago












            $begingroup$
            I'm glad I could help.
            $endgroup$
            – José Carlos Santos
            3 hours ago




            $begingroup$
            I'm glad I could help.
            $endgroup$
            – José Carlos Santos
            3 hours ago











            3












            $begingroup$

            The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



            Hence Santos' example was the standard example of a non-separable metric space.






            share|cite|improve this answer









            $endgroup$


















              3












              $begingroup$

              The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



              Hence Santos' example was the standard example of a non-separable metric space.






              share|cite|improve this answer









              $endgroup$
















                3












                3








                3





                $begingroup$

                The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



                Hence Santos' example was the standard example of a non-separable metric space.






                share|cite|improve this answer









                $endgroup$



                The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



                Hence Santos' example was the standard example of a non-separable metric space.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Henno BrandsmaHenno Brandsma

                109k347114




                109k347114






























                    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%2f3106782%2fif-a-collection-of-closed-sets-of-arbitrary-cardinality-in-a-metric-space-has-em%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

                    Усть-Каменогорск

                    Image
                    У этого термина существуют и другие значения, см. Усть-Каменогорск. Город Усть-Каменогорск каз. Өскемен Флаг Герб 49°57′ с. ш. 82°37′ в. д. H G Я O L Страна Казахстан   Казахстан Статус областной центр Область Восточно-Казахстанская Аким Жаксылык Омар [1] История и география Основан 1720 Город с 1868 Площадь 303,15 [2] км² Высота центра 283 ± 1 м Тип климата резко-континентальный Часовой пояс UTC+6 Население Население 329 090 [3]  человек ( 2018 ) Национальности русские 53,56 % казахи 42,99 % немцы 0,82 % украинцы 0,57 % татары 0,79 % белорусы 0,14 % азербайджанцы 0,23 % корейцы 0,20 % прочие 1,08 % -1 шт [4] Этнохороним усть-каменогорцы, усть-каменогорец Цифровые идентификаторы Телефонный код +7 (7232) Почтовый индекс F0***** [5] Автомобильный код 16 (ранее F) oskemen.kz Усть-Каменогорск Усть-Каменогорск ...
                    Read more

                    Халкинская богословская школа

                    Image
                    Халкинская богословская школа тур. Rum Ortodoks Ruhban Okulu греч. Ιερά Θεολογική Σχολή της Χάλκης Главное здание Халкинской школы Международное название Theological School of Halki Год основания 1844 Конфессия православие Церковь Константинопольская Ректор Елпидифор (Ламбриниадис) Расположение о. Хейбелиада, Турция Юридический адрес Ümit tepesi No 5, 34973 Heybeliada, Istanbul, Turkey Сайт www.greece.org  [[commons:Category:Halki seminary|Халкинская богословская школа тур. Rum Ortodoks Ruhban Okulu греч. Ιερά Θεολογική Σχολή της Χάλκης на Викискладе]] Ха́лкинская богосло́вская шко́ла (или Духовная семинария на Халки , тур. Rum Ortodoks Ruhban Okulu , греч. Ιερά Θεολογική Σχολή της Χάλκης ) — недействующее среднее специальное учебное заведение Константинопольской православной церкви, расположенное на территории Свято-Троицкого мужского монастыря на вершине холма Надежды (тур. Ümit Tepesi ) острова Хейбелиада, в Турции....
                    Read more

                    Высокополье (Харьковская область)

                    Image
                    У этого термина существуют и другие значения, см. Высокополье. Село Высокополье укр. Високопілля Герб 49°55′07″ с. ш. 35°25′13″ в. д. H G Я O L Страна Украина   Украина Область Харьковская Район Валковский Сельский совет Высокопольский История и география Основан 1670 Прежние названия Хмелевое Площадь 55,56 км² Часовой пояс UTC+2, летом UTC+3 Население Население 1957 человек ( 2001 ) Плотность 35,220 чел./км² Цифровые идентификаторы Телефонный код +380 5753 Почтовый индекс 63020 Автомобильный код AX, КХ / 21 КОАТУУ 6321281501 Высокополье Высокополье Высокополье  Аудио, фото и видео на Викискладе Высокопо́лье (укр. Високопілля ) — село, Высокопольский сельский совет, Валковский район, Харьковская область, Украина. Код КОАТУУ — 6321281501. Население по переписи 2001 г. составляет 1957 (866/1091 м/ж) человек. Является административным центр...
                    Read more