Given set U is first countable or not?











up vote
-1
down vote

favorite












In $mathbb{R}$ with usual topology ,the set $U ={ x in mathbb{R} : -1le x le 1 , ,x neq 0}$ is



Choose the correct statement



$a)$ Neither hausdorff nor First counatble



$b)$ Hausdorff



$c)$ First countable



$d)$both hausdorff and first countable



My attempt :set $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint set, From this i can concnclude that $U$ is hausdorff



Im confusing that it is First countable or not ?



Any hints/solution will be appreciated



thanks u






general-topology separation-axioms first-countable







share|cite|improve this question




























    up vote
    -1
    down vote

    favorite












    In $mathbb{R}$ with usual topology ,the set $U ={ x in mathbb{R} : -1le x le 1 , ,x neq 0}$ is



    Choose the correct statement



    $a)$ Neither hausdorff nor First counatble



    $b)$ Hausdorff



    $c)$ First countable



    $d)$both hausdorff and first countable



    My attempt :set $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint set, From this i can concnclude that $U$ is hausdorff



    Im confusing that it is First countable or not ?



    Any hints/solution will be appreciated



    thanks u






    general-topology separation-axioms first-countable







    share|cite|improve this question


























      up vote
      -1
      down vote

      favorite









      up vote
      -1
      down vote

      favorite











      In $mathbb{R}$ with usual topology ,the set $U ={ x in mathbb{R} : -1le x le 1 , ,x neq 0}$ is



      Choose the correct statement



      $a)$ Neither hausdorff nor First counatble



      $b)$ Hausdorff



      $c)$ First countable



      $d)$both hausdorff and first countable



      My attempt :set $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint set, From this i can concnclude that $U$ is hausdorff



      Im confusing that it is First countable or not ?



      Any hints/solution will be appreciated



      thanks u






      general-topology separation-axioms first-countable







      share|cite|improve this question















      In $mathbb{R}$ with usual topology ,the set $U ={ x in mathbb{R} : -1le x le 1 , ,x neq 0}$ is



      Choose the correct statement



      $a)$ Neither hausdorff nor First counatble



      $b)$ Hausdorff



      $c)$ First countable



      $d)$both hausdorff and first countable



      My attempt :set $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint set, From this i can concnclude that $U$ is hausdorff



      Im confusing that it is First countable or not ?



      Any hints/solution will be appreciated



      thanks u






      general-topology separation-axioms first-countable




      general-topology separation-axioms first-countable






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 22 at 13:43









      José Carlos Santos

      146k22117217




      146k22117217










      asked Nov 22 at 13:39









      jasmine

      1,505416




      1,505416






















          3 Answers
          3






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          The usual topology is induced by a metric and every metric space is first-countable.






          share|cite|improve this answer




























            up vote
            2
            down vote













            $mathbb{R}$ with usual topology is also a metric space. So $mathbb{R}$ is first countable. Hence any subspace is also first countable.






            share|cite|improve this answer

















            • 1




              To check Hausdorffness use definition.
              – Offlaw
              Nov 22 at 13:47


















            up vote
            1
            down vote














            $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint
            set, From this i can concnclude that $U$ is hausdorff




            The fact that $U$ can be written as a union of two disjoint sets has nothing to do with the set being Hausforff or not.



            For the first countable property, google is your friend.






            share|cite|improve this answer





















              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',
              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%2f3009148%2fgiven-set-u-is-first-countable-or-not%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              3 Answers
              3






              active

              oldest

              votes








              3 Answers
              3






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              1
              down vote



              accepted










              The usual topology is induced by a metric and every metric space is first-countable.






              share|cite|improve this answer

























                up vote
                1
                down vote



                accepted










                The usual topology is induced by a metric and every metric space is first-countable.






                share|cite|improve this answer























                  up vote
                  1
                  down vote



                  accepted







                  up vote
                  1
                  down vote



                  accepted






                  The usual topology is induced by a metric and every metric space is first-countable.






                  share|cite|improve this answer












                  The usual topology is induced by a metric and every metric space is first-countable.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 22 at 13:41









                  José Carlos Santos

                  146k22117217




                  146k22117217






















                      up vote
                      2
                      down vote













                      $mathbb{R}$ with usual topology is also a metric space. So $mathbb{R}$ is first countable. Hence any subspace is also first countable.






                      share|cite|improve this answer

















                      • 1




                        To check Hausdorffness use definition.
                        – Offlaw
                        Nov 22 at 13:47















                      up vote
                      2
                      down vote













                      $mathbb{R}$ with usual topology is also a metric space. So $mathbb{R}$ is first countable. Hence any subspace is also first countable.






                      share|cite|improve this answer

















                      • 1




                        To check Hausdorffness use definition.
                        – Offlaw
                        Nov 22 at 13:47













                      up vote
                      2
                      down vote










                      up vote
                      2
                      down vote









                      $mathbb{R}$ with usual topology is also a metric space. So $mathbb{R}$ is first countable. Hence any subspace is also first countable.






                      share|cite|improve this answer












                      $mathbb{R}$ with usual topology is also a metric space. So $mathbb{R}$ is first countable. Hence any subspace is also first countable.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Nov 22 at 13:43









                      Offlaw

                      2649




                      2649








                      • 1




                        To check Hausdorffness use definition.
                        – Offlaw
                        Nov 22 at 13:47














                      • 1




                        To check Hausdorffness use definition.
                        – Offlaw
                        Nov 22 at 13:47








                      1




                      1




                      To check Hausdorffness use definition.
                      – Offlaw
                      Nov 22 at 13:47




                      To check Hausdorffness use definition.
                      – Offlaw
                      Nov 22 at 13:47










                      up vote
                      1
                      down vote














                      $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint
                      set, From this i can concnclude that $U$ is hausdorff




                      The fact that $U$ can be written as a union of two disjoint sets has nothing to do with the set being Hausforff or not.



                      For the first countable property, google is your friend.






                      share|cite|improve this answer

























                        up vote
                        1
                        down vote














                        $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint
                        set, From this i can concnclude that $U$ is hausdorff




                        The fact that $U$ can be written as a union of two disjoint sets has nothing to do with the set being Hausforff or not.



                        For the first countable property, google is your friend.






                        share|cite|improve this answer























                          up vote
                          1
                          down vote










                          up vote
                          1
                          down vote










                          $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint
                          set, From this i can concnclude that $U$ is hausdorff




                          The fact that $U$ can be written as a union of two disjoint sets has nothing to do with the set being Hausforff or not.



                          For the first countable property, google is your friend.






                          share|cite|improve this answer













                          $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint
                          set, From this i can concnclude that $U$ is hausdorff




                          The fact that $U$ can be written as a union of two disjoint sets has nothing to do with the set being Hausforff or not.



                          For the first countable property, google is your friend.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Nov 22 at 13:42









                          5xum

                          89.4k393161




                          89.4k393161






























                              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.





                              Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                              Please pay close attention to the following guidance:


                              • 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.


                              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%2f3009148%2fgiven-set-u-is-first-countable-or-not%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