What is an infinite subset of a compact set?












2












$begingroup$


I am attempting to work on the following proof:



If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.



I know that that this proof has been answered here already, but I am more interested in understanding the statement itself. I am just struggling to comprehend what $E$ is in this question... an infinite subset of a compact set. For some reason, I just cannot visualize this. If anyone out there is able to help me see through the fog, or even provide a specific example for me to think about, I would greatly appreciate it.






general-topology







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    I am attempting to work on the following proof:



    If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.



    I know that that this proof has been answered here already, but I am more interested in understanding the statement itself. I am just struggling to comprehend what $E$ is in this question... an infinite subset of a compact set. For some reason, I just cannot visualize this. If anyone out there is able to help me see through the fog, or even provide a specific example for me to think about, I would greatly appreciate it.






    general-topology







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I am attempting to work on the following proof:



      If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.



      I know that that this proof has been answered here already, but I am more interested in understanding the statement itself. I am just struggling to comprehend what $E$ is in this question... an infinite subset of a compact set. For some reason, I just cannot visualize this. If anyone out there is able to help me see through the fog, or even provide a specific example for me to think about, I would greatly appreciate it.






      general-topology







      share|cite|improve this question









      $endgroup$




      I am attempting to work on the following proof:



      If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.



      I know that that this proof has been answered here already, but I am more interested in understanding the statement itself. I am just struggling to comprehend what $E$ is in this question... an infinite subset of a compact set. For some reason, I just cannot visualize this. If anyone out there is able to help me see through the fog, or even provide a specific example for me to think about, I would greatly appreciate it.






      general-topology




      general-topology






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 7 '18 at 5:05









      automattikautomattik

      406




      406






















          2 Answers
          2






          active

          oldest

          votes


















          3












          $begingroup$

          There are many possibilities. Take $(0,1)subseteq [0,1]$ or the circle embedded as the equator of the sphere: $S^1subseteq S^2$.



          If you are familiar with it, take the middle thirds cantor set $Csubseteq [0,1]$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
            $endgroup$
            – automattik
            Dec 8 '18 at 5:38








          • 1




            $begingroup$
            Yes that's also correct.
            $endgroup$
            – Antonios-Alexandros Robotis
            Dec 8 '18 at 5:39



















          0












          $begingroup$

          There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $I$ for example, you can pick the sequence $1,frac{1}{2},...,1/n...$ as an infinite subset, or any open(or closed) interval contained in $I$.



          Although not mentioned in the description, a proof of the theorem can be found here:
          Infinite subset of a compact set






          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%2f3029487%2fwhat-is-an-infinite-subset-of-a-compact-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









            3












            $begingroup$

            There are many possibilities. Take $(0,1)subseteq [0,1]$ or the circle embedded as the equator of the sphere: $S^1subseteq S^2$.



            If you are familiar with it, take the middle thirds cantor set $Csubseteq [0,1]$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
              $endgroup$
              – automattik
              Dec 8 '18 at 5:38








            • 1




              $begingroup$
              Yes that's also correct.
              $endgroup$
              – Antonios-Alexandros Robotis
              Dec 8 '18 at 5:39
















            3












            $begingroup$

            There are many possibilities. Take $(0,1)subseteq [0,1]$ or the circle embedded as the equator of the sphere: $S^1subseteq S^2$.



            If you are familiar with it, take the middle thirds cantor set $Csubseteq [0,1]$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
              $endgroup$
              – automattik
              Dec 8 '18 at 5:38








            • 1




              $begingroup$
              Yes that's also correct.
              $endgroup$
              – Antonios-Alexandros Robotis
              Dec 8 '18 at 5:39














            3












            3








            3





            $begingroup$

            There are many possibilities. Take $(0,1)subseteq [0,1]$ or the circle embedded as the equator of the sphere: $S^1subseteq S^2$.



            If you are familiar with it, take the middle thirds cantor set $Csubseteq [0,1]$.






            share|cite|improve this answer









            $endgroup$



            There are many possibilities. Take $(0,1)subseteq [0,1]$ or the circle embedded as the equator of the sphere: $S^1subseteq S^2$.



            If you are familiar with it, take the middle thirds cantor set $Csubseteq [0,1]$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 7 '18 at 5:08









            Antonios-Alexandros RobotisAntonios-Alexandros Robotis

            9,91741640




            9,91741640












            • $begingroup$
              Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
              $endgroup$
              – automattik
              Dec 8 '18 at 5:38








            • 1




              $begingroup$
              Yes that's also correct.
              $endgroup$
              – Antonios-Alexandros Robotis
              Dec 8 '18 at 5:39


















            • $begingroup$
              Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
              $endgroup$
              – automattik
              Dec 8 '18 at 5:38








            • 1




              $begingroup$
              Yes that's also correct.
              $endgroup$
              – Antonios-Alexandros Robotis
              Dec 8 '18 at 5:39
















            $begingroup$
            Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
            $endgroup$
            – automattik
            Dec 8 '18 at 5:38






            $begingroup$
            Thank you for the examples. Would it also be correct to say that $[0,1] subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well.
            $endgroup$
            – automattik
            Dec 8 '18 at 5:38






            1




            1




            $begingroup$
            Yes that's also correct.
            $endgroup$
            – Antonios-Alexandros Robotis
            Dec 8 '18 at 5:39




            $begingroup$
            Yes that's also correct.
            $endgroup$
            – Antonios-Alexandros Robotis
            Dec 8 '18 at 5:39











            0












            $begingroup$

            There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $I$ for example, you can pick the sequence $1,frac{1}{2},...,1/n...$ as an infinite subset, or any open(or closed) interval contained in $I$.



            Although not mentioned in the description, a proof of the theorem can be found here:
            Infinite subset of a compact set






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $I$ for example, you can pick the sequence $1,frac{1}{2},...,1/n...$ as an infinite subset, or any open(or closed) interval contained in $I$.



              Although not mentioned in the description, a proof of the theorem can be found here:
              Infinite subset of a compact set






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $I$ for example, you can pick the sequence $1,frac{1}{2},...,1/n...$ as an infinite subset, or any open(or closed) interval contained in $I$.



                Although not mentioned in the description, a proof of the theorem can be found here:
                Infinite subset of a compact set






                share|cite|improve this answer









                $endgroup$



                There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $I$ for example, you can pick the sequence $1,frac{1}{2},...,1/n...$ as an infinite subset, or any open(or closed) interval contained in $I$.



                Although not mentioned in the description, a proof of the theorem can be found here:
                Infinite subset of a compact set







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 7 '18 at 5:22









                William SunWilliam Sun

                471111




                471111






























                    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%2f3029487%2fwhat-is-an-infinite-subset-of-a-compact-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

                    Ellipse (mathématiques)

                    Image
                    Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
                    Read more

                    Quarter-circle Tiles

                    Image
                    up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
                    Read more

                    Mont Emei

                    Image
                    Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
                    Read more