a relationship between local compactness and closure












3












$begingroup$


Suppose that $X$ is a Hausdorff locally compact space and $S$ a subset of $X$. Let $xin X$ and suppose that every compact neighborhood of $x$ intersects $S$. Does it follow that $x$ lies in the closure of $S$?










share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    Suppose that $X$ is a Hausdorff locally compact space and $S$ a subset of $X$. Let $xin X$ and suppose that every compact neighborhood of $x$ intersects $S$. Does it follow that $x$ lies in the closure of $S$?










    share|cite|improve this question









    $endgroup$















      3












      3








      3


      1



      $begingroup$


      Suppose that $X$ is a Hausdorff locally compact space and $S$ a subset of $X$. Let $xin X$ and suppose that every compact neighborhood of $x$ intersects $S$. Does it follow that $x$ lies in the closure of $S$?










      share|cite|improve this question









      $endgroup$




      Suppose that $X$ is a Hausdorff locally compact space and $S$ a subset of $X$. Let $xin X$ and suppose that every compact neighborhood of $x$ intersects $S$. Does it follow that $x$ lies in the closure of $S$?







      general-topology






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 12 hours ago









      User12239User12239

      367216




      367216






















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          Yes, it is true. Since $X$ is locally compact and Hausdorff, given a neighborhood $U$ of $x$ in $X$, we can find a neighborhood $V$ such that $overline V$ is compact and $overline Vsubseteq U$. As every compact neighborhood of $x$ intersects with $S$, $overline Vcap Sneq emptyset$ $implies Ucap Sneq emptyset$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
            $endgroup$
            – User12239
            12 hours ago










          • $begingroup$
            You can find a proof in 'Topology' by Munkres.
            $endgroup$
            – Thomas Shelby
            12 hours ago










          • $begingroup$
            Also see this.
            $endgroup$
            – Thomas Shelby
            12 hours ago






          • 1




            $begingroup$
            Thanks I’m looking them up
            $endgroup$
            – User12239
            12 hours ago



















          2












          $begingroup$

          In this case ($X$ being locally compact Hausdorff) for every point $x$ of $X$, every neighbourhood of $x$ contains a compact neighbourhood of $x$ - see.



          Then any neighbourhood of $x$ contains a compact neighbourhood which intersects $S$, so $xin bar{S}$






          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%2f3177090%2fa-relationship-between-local-compactness-and-closure%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









            2












            $begingroup$

            Yes, it is true. Since $X$ is locally compact and Hausdorff, given a neighborhood $U$ of $x$ in $X$, we can find a neighborhood $V$ such that $overline V$ is compact and $overline Vsubseteq U$. As every compact neighborhood of $x$ intersects with $S$, $overline Vcap Sneq emptyset$ $implies Ucap Sneq emptyset$.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
              $endgroup$
              – User12239
              12 hours ago










            • $begingroup$
              You can find a proof in 'Topology' by Munkres.
              $endgroup$
              – Thomas Shelby
              12 hours ago










            • $begingroup$
              Also see this.
              $endgroup$
              – Thomas Shelby
              12 hours ago






            • 1




              $begingroup$
              Thanks I’m looking them up
              $endgroup$
              – User12239
              12 hours ago
















            2












            $begingroup$

            Yes, it is true. Since $X$ is locally compact and Hausdorff, given a neighborhood $U$ of $x$ in $X$, we can find a neighborhood $V$ such that $overline V$ is compact and $overline Vsubseteq U$. As every compact neighborhood of $x$ intersects with $S$, $overline Vcap Sneq emptyset$ $implies Ucap Sneq emptyset$.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
              $endgroup$
              – User12239
              12 hours ago










            • $begingroup$
              You can find a proof in 'Topology' by Munkres.
              $endgroup$
              – Thomas Shelby
              12 hours ago










            • $begingroup$
              Also see this.
              $endgroup$
              – Thomas Shelby
              12 hours ago






            • 1




              $begingroup$
              Thanks I’m looking them up
              $endgroup$
              – User12239
              12 hours ago














            2












            2








            2





            $begingroup$

            Yes, it is true. Since $X$ is locally compact and Hausdorff, given a neighborhood $U$ of $x$ in $X$, we can find a neighborhood $V$ such that $overline V$ is compact and $overline Vsubseteq U$. As every compact neighborhood of $x$ intersects with $S$, $overline Vcap Sneq emptyset$ $implies Ucap Sneq emptyset$.






            share|cite|improve this answer











            $endgroup$



            Yes, it is true. Since $X$ is locally compact and Hausdorff, given a neighborhood $U$ of $x$ in $X$, we can find a neighborhood $V$ such that $overline V$ is compact and $overline Vsubseteq U$. As every compact neighborhood of $x$ intersects with $S$, $overline Vcap Sneq emptyset$ $implies Ucap Sneq emptyset$.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 12 hours ago

























            answered 12 hours ago









            Thomas ShelbyThomas Shelby

            4,7362727




            4,7362727












            • $begingroup$
              Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
              $endgroup$
              – User12239
              12 hours ago










            • $begingroup$
              You can find a proof in 'Topology' by Munkres.
              $endgroup$
              – Thomas Shelby
              12 hours ago










            • $begingroup$
              Also see this.
              $endgroup$
              – Thomas Shelby
              12 hours ago






            • 1




              $begingroup$
              Thanks I’m looking them up
              $endgroup$
              – User12239
              12 hours ago


















            • $begingroup$
              Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
              $endgroup$
              – User12239
              12 hours ago










            • $begingroup$
              You can find a proof in 'Topology' by Munkres.
              $endgroup$
              – Thomas Shelby
              12 hours ago










            • $begingroup$
              Also see this.
              $endgroup$
              – Thomas Shelby
              12 hours ago






            • 1




              $begingroup$
              Thanks I’m looking them up
              $endgroup$
              – User12239
              12 hours ago
















            $begingroup$
            Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
            $endgroup$
            – User12239
            12 hours ago




            $begingroup$
            Yes I was doubting if every neighborhood has a compact neighborhood so I need to prove this fact now
            $endgroup$
            – User12239
            12 hours ago












            $begingroup$
            You can find a proof in 'Topology' by Munkres.
            $endgroup$
            – Thomas Shelby
            12 hours ago




            $begingroup$
            You can find a proof in 'Topology' by Munkres.
            $endgroup$
            – Thomas Shelby
            12 hours ago












            $begingroup$
            Also see this.
            $endgroup$
            – Thomas Shelby
            12 hours ago




            $begingroup$
            Also see this.
            $endgroup$
            – Thomas Shelby
            12 hours ago




            1




            1




            $begingroup$
            Thanks I’m looking them up
            $endgroup$
            – User12239
            12 hours ago




            $begingroup$
            Thanks I’m looking them up
            $endgroup$
            – User12239
            12 hours ago











            2












            $begingroup$

            In this case ($X$ being locally compact Hausdorff) for every point $x$ of $X$, every neighbourhood of $x$ contains a compact neighbourhood of $x$ - see.



            Then any neighbourhood of $x$ contains a compact neighbourhood which intersects $S$, so $xin bar{S}$






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              In this case ($X$ being locally compact Hausdorff) for every point $x$ of $X$, every neighbourhood of $x$ contains a compact neighbourhood of $x$ - see.



              Then any neighbourhood of $x$ contains a compact neighbourhood which intersects $S$, so $xin bar{S}$






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                In this case ($X$ being locally compact Hausdorff) for every point $x$ of $X$, every neighbourhood of $x$ contains a compact neighbourhood of $x$ - see.



                Then any neighbourhood of $x$ contains a compact neighbourhood which intersects $S$, so $xin bar{S}$






                share|cite|improve this answer









                $endgroup$



                In this case ($X$ being locally compact Hausdorff) for every point $x$ of $X$, every neighbourhood of $x$ contains a compact neighbourhood of $x$ - see.



                Then any neighbourhood of $x$ contains a compact neighbourhood which intersects $S$, so $xin bar{S}$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 12 hours ago









                MaksimMaksim

                1,00719




                1,00719






























                    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%2f3177090%2fa-relationship-between-local-compactness-and-closure%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

                    Knooppunt Holsloot

                    Altaar (religie)

                    Gregoriusmis