Prove that $A$ is homeomorphic to $S^1times[1,2]$ where $A$ and $S^1$ are defined below:












-1














let A = ${(x,y) in mathbb R^2 : 1 le sqrt{x^2+y^2} le 2 }$ . Prove that $A$ is homeomorphic to $S^1times[1,2]$ where $S^1 = {(x,y) in mathbb R^2 : x^2+y^2 =1 }$










share|cite|improve this question





























    -1














    let A = ${(x,y) in mathbb R^2 : 1 le sqrt{x^2+y^2} le 2 }$ . Prove that $A$ is homeomorphic to $S^1times[1,2]$ where $S^1 = {(x,y) in mathbb R^2 : x^2+y^2 =1 }$










    share|cite|improve this question



























      -1












      -1








      -1







      let A = ${(x,y) in mathbb R^2 : 1 le sqrt{x^2+y^2} le 2 }$ . Prove that $A$ is homeomorphic to $S^1times[1,2]$ where $S^1 = {(x,y) in mathbb R^2 : x^2+y^2 =1 }$










      share|cite|improve this question















      let A = ${(x,y) in mathbb R^2 : 1 le sqrt{x^2+y^2} le 2 }$ . Prove that $A$ is homeomorphic to $S^1times[1,2]$ where $S^1 = {(x,y) in mathbb R^2 : x^2+y^2 =1 }$







      general-topology






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 24 at 13:14









      Tianlalu

      3,02021038




      3,02021038










      asked Nov 24 at 13:03









      Joy

      12




      12






















          1 Answer
          1






          active

          oldest

          votes


















          0














          Notice that $A={(rcos theta,rsin theta): 1≤r≤2,0≤theta<2pi}$. Therefore required homeomorphism is $f:Arightarrow S^1×[1,2]$ is given by $f(rcos theta,rsin theta)=((costheta,sintheta),r)$.To show $f$ is homeomorphism first show it is bijective and continuous. Now $A$ is compact and $S^1×[1,2]$ is hausdorff i.e. $f^{-1}$ is also continuous.






          share|cite|improve this answer























          • I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
            – UserS
            Nov 24 at 13:26











          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%2f3011528%2fprove-that-a-is-homeomorphic-to-s1-times1-2-where-a-and-s1-are-defin%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          0














          Notice that $A={(rcos theta,rsin theta): 1≤r≤2,0≤theta<2pi}$. Therefore required homeomorphism is $f:Arightarrow S^1×[1,2]$ is given by $f(rcos theta,rsin theta)=((costheta,sintheta),r)$.To show $f$ is homeomorphism first show it is bijective and continuous. Now $A$ is compact and $S^1×[1,2]$ is hausdorff i.e. $f^{-1}$ is also continuous.






          share|cite|improve this answer























          • I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
            – UserS
            Nov 24 at 13:26
















          0














          Notice that $A={(rcos theta,rsin theta): 1≤r≤2,0≤theta<2pi}$. Therefore required homeomorphism is $f:Arightarrow S^1×[1,2]$ is given by $f(rcos theta,rsin theta)=((costheta,sintheta),r)$.To show $f$ is homeomorphism first show it is bijective and continuous. Now $A$ is compact and $S^1×[1,2]$ is hausdorff i.e. $f^{-1}$ is also continuous.






          share|cite|improve this answer























          • I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
            – UserS
            Nov 24 at 13:26














          0












          0








          0






          Notice that $A={(rcos theta,rsin theta): 1≤r≤2,0≤theta<2pi}$. Therefore required homeomorphism is $f:Arightarrow S^1×[1,2]$ is given by $f(rcos theta,rsin theta)=((costheta,sintheta),r)$.To show $f$ is homeomorphism first show it is bijective and continuous. Now $A$ is compact and $S^1×[1,2]$ is hausdorff i.e. $f^{-1}$ is also continuous.






          share|cite|improve this answer














          Notice that $A={(rcos theta,rsin theta): 1≤r≤2,0≤theta<2pi}$. Therefore required homeomorphism is $f:Arightarrow S^1×[1,2]$ is given by $f(rcos theta,rsin theta)=((costheta,sintheta),r)$.To show $f$ is homeomorphism first show it is bijective and continuous. Now $A$ is compact and $S^1×[1,2]$ is hausdorff i.e. $f^{-1}$ is also continuous.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 24 at 13:18

























          answered Nov 24 at 13:12









          UserS

          1,5371112




          1,5371112












          • I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
            – UserS
            Nov 24 at 13:26


















          • I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
            – UserS
            Nov 24 at 13:26
















          I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
          – UserS
          Nov 24 at 13:26




          I think you know that to show a function defined on a metric space is continuous it is enough to show that $x_nrightarrow ximplies f(x_n)rightarrow f(x)$.
          – UserS
          Nov 24 at 13:26


















          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%2f3011528%2fprove-that-a-is-homeomorphic-to-s1-times1-2-where-a-and-s1-are-defin%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)

          Quarter-circle Tiles

          Mont Emei