Existence of an homeomorphic between [0,1] to X × YIs $[0,1]^omega$ homeomorphic to $D^omega$?Prove rigorously that for two points $x, y in M$, the spaces $M backslash x$ and $M backslash y$ are homeomorphic.Does the Stone-Čech compactification respect subspaces?How to show that $[0,1)$ and $(0,1]$ are or aren't homeomorphic with induced $mathcalT_l$ topologies?Are X and Y homeomorphic?Two topological spaces which imbed in each other and are quotients of each other but not homeomorphic?An example of non-homeomorphic surfaces, $S_1, S_2$, such that $S_1 times [0,1]$ is homeomorphic to $S_2 times [0,1]$construct two non-homeomorphic topological spacesComplete+bounded homeomorphic to incomplete+unboundeddistinction between homeomorphic topological spaces

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Existence of an homeomorphic between [0,1] to X × Y


Is $[0,1]^omega$ homeomorphic to $D^omega$?Prove rigorously that for two points $x, y in M$, the spaces $M backslash x$ and $M backslash y$ are homeomorphic.Does the Stone-Čech compactification respect subspaces?How to show that $[0,1)$ and $(0,1]$ are or aren't homeomorphic with induced $mathcalT_l$ topologies?Are X and Y homeomorphic?Two topological spaces which imbed in each other and are quotients of each other but not homeomorphic?An example of non-homeomorphic surfaces, $S_1, S_2$, such that $S_1 times [0,1]$ is homeomorphic to $S_2 times [0,1]$construct two non-homeomorphic topological spacesComplete+bounded homeomorphic to incomplete+unboundeddistinction between homeomorphic topological spaces






.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;








3












$begingroup$


I'm doing a practice exam questions and am stuck at this question:




Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?




I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./



Thanks in advance!










share|cite|improve this question









$endgroup$


















    3












    $begingroup$


    I'm doing a practice exam questions and am stuck at this question:




    Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?




    I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./



    Thanks in advance!










    share|cite|improve this question









    $endgroup$














      3












      3








      3





      $begingroup$


      I'm doing a practice exam questions and am stuck at this question:




      Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?




      I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./



      Thanks in advance!










      share|cite|improve this question









      $endgroup$




      I'm doing a practice exam questions and am stuck at this question:




      Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?




      I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./



      Thanks in advance!







      general-topology






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jun 11 at 8:25









      WorriorWorrior

      354




      354




















          2 Answers
          2






          active

          oldest

          votes


















          7












          $begingroup$

          You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.



          Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?



          (Hint: make a picture and try to connect two arbitrary points with a path.)






          share|cite|improve this answer









          $endgroup$




















            4












            $begingroup$

            Hints: Prove that:



            • If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.

            • If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.





            share|cite|improve this answer









            $endgroup$













              Your Answer








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              2 Answers
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              active

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              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              7












              $begingroup$

              You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.



              Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?



              (Hint: make a picture and try to connect two arbitrary points with a path.)






              share|cite|improve this answer









              $endgroup$

















                7












                $begingroup$

                You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.



                Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?



                (Hint: make a picture and try to connect two arbitrary points with a path.)






                share|cite|improve this answer









                $endgroup$















                  7












                  7








                  7





                  $begingroup$

                  You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.



                  Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?



                  (Hint: make a picture and try to connect two arbitrary points with a path.)






                  share|cite|improve this answer









                  $endgroup$



                  You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.



                  Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?



                  (Hint: make a picture and try to connect two arbitrary points with a path.)







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jun 11 at 8:39









                  PedroPedro

                  3,0791722




                  3,0791722























                      4












                      $begingroup$

                      Hints: Prove that:



                      • If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.

                      • If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.





                      share|cite|improve this answer









                      $endgroup$

















                        4












                        $begingroup$

                        Hints: Prove that:



                        • If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.

                        • If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.





                        share|cite|improve this answer









                        $endgroup$















                          4












                          4








                          4





                          $begingroup$

                          Hints: Prove that:



                          • If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.

                          • If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.





                          share|cite|improve this answer









                          $endgroup$



                          Hints: Prove that:



                          • If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.

                          • If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.






                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Jun 11 at 8:32









                          José Carlos SantosJosé Carlos Santos

                          193k24148266




                          193k24148266



























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