What is the mathematical word to describe when two objects can be transformed to each other like Klein bottle...












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In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?






general-topology terminology







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  • 2




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    Homeomorphic?
    $endgroup$
    – DreamConspiracy
    Dec 7 '18 at 16:46












  • $begingroup$
    @DreamConspiracy yes, thank you.
    $endgroup$
    – katerine
    Dec 7 '18 at 16:48






  • 3




    $begingroup$
    @katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
    $endgroup$
    – Randall
    Dec 7 '18 at 16:51


















3












$begingroup$


In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?






general-topology terminology







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Homeomorphic?
    $endgroup$
    – DreamConspiracy
    Dec 7 '18 at 16:46












  • $begingroup$
    @DreamConspiracy yes, thank you.
    $endgroup$
    – katerine
    Dec 7 '18 at 16:48






  • 3




    $begingroup$
    @katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
    $endgroup$
    – Randall
    Dec 7 '18 at 16:51
















3












3








3





$begingroup$


In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?






general-topology terminology







share|cite|improve this question











$endgroup$




In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?






general-topology terminology




general-topology terminology






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 17:19









Brahadeesh

6,50942363




6,50942363










asked Dec 7 '18 at 16:43









katerinekaterine

326




326








  • 2




    $begingroup$
    Homeomorphic?
    $endgroup$
    – DreamConspiracy
    Dec 7 '18 at 16:46












  • $begingroup$
    @DreamConspiracy yes, thank you.
    $endgroup$
    – katerine
    Dec 7 '18 at 16:48






  • 3




    $begingroup$
    @katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
    $endgroup$
    – Randall
    Dec 7 '18 at 16:51
















  • 2




    $begingroup$
    Homeomorphic?
    $endgroup$
    – DreamConspiracy
    Dec 7 '18 at 16:46












  • $begingroup$
    @DreamConspiracy yes, thank you.
    $endgroup$
    – katerine
    Dec 7 '18 at 16:48






  • 3




    $begingroup$
    @katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
    $endgroup$
    – Randall
    Dec 7 '18 at 16:51










2




2




$begingroup$
Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46






$begingroup$
Homeomorphic?
$endgroup$
– DreamConspiracy
Dec 7 '18 at 16:46














$begingroup$
@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48




$begingroup$
@DreamConspiracy yes, thank you.
$endgroup$
– katerine
Dec 7 '18 at 16:48




3




3




$begingroup$
@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51






$begingroup$
@katerine To be clear, a torus and the Klein bottle are NOT homeomorphic.
$endgroup$
– Randall
Dec 7 '18 at 16:51












2 Answers
2






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4












$begingroup$

There are a few notions depending on what type of equivalence you want to consider.



• A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.



• Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.



• A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.



Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.



There are also other notions of equivalence like isometry, etc.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.






    share|cite|improve this answer











    $endgroup$









    • 2




      $begingroup$
      Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
      $endgroup$
      – Noah Schweber
      Dec 7 '18 at 17:19












    • $begingroup$
      @NoahSchweber brain fart, thank you
      $endgroup$
      – DreamConspiracy
      Dec 7 '18 at 17:22











    Your Answer





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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

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    4












    $begingroup$

    There are a few notions depending on what type of equivalence you want to consider.



    • A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.



    • Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.



    • A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.



    Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.



    There are also other notions of equivalence like isometry, etc.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      There are a few notions depending on what type of equivalence you want to consider.



      • A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.



      • Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.



      • A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.



      Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.



      There are also other notions of equivalence like isometry, etc.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        There are a few notions depending on what type of equivalence you want to consider.



        • A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.



        • Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.



        • A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.



        Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.



        There are also other notions of equivalence like isometry, etc.






        share|cite|improve this answer









        $endgroup$



        There are a few notions depending on what type of equivalence you want to consider.



        • A diffeomorphism of two manifolds $M$ and $N$ is a smooth bijection $f:Mto N$ with smooth inverse.



        • Homeomorphism of two topological spaces $X$ and $Y$ is when there exists a map $f:Xto Y$ which is a continuous bijection with continuous inverse.



        • A homotopy equivalence of spaces is a pair of maps $f:Xto Y$ and $g:Yto X$ with $gcirc fsimeq operatorname{Id}_X$ and $fcirc g simeq operatorname{Id}_Y$.



        Diffeomorphism $implies$ Homeomorphism $implies$ Homotopy Equivalence.



        There are also other notions of equivalence like isometry, etc.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 7 '18 at 16:53









        Antonios-Alexandros RobotisAntonios-Alexandros Robotis

        10.5k41741




        10.5k41741























            2












            $begingroup$

            Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.






            share|cite|improve this answer











            $endgroup$









            • 2




              $begingroup$
              Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
              $endgroup$
              – Noah Schweber
              Dec 7 '18 at 17:19












            • $begingroup$
              @NoahSchweber brain fart, thank you
              $endgroup$
              – DreamConspiracy
              Dec 7 '18 at 17:22
















            2












            $begingroup$

            Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.






            share|cite|improve this answer











            $endgroup$









            • 2




              $begingroup$
              Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
              $endgroup$
              – Noah Schweber
              Dec 7 '18 at 17:19












            • $begingroup$
              @NoahSchweber brain fart, thank you
              $endgroup$
              – DreamConspiracy
              Dec 7 '18 at 17:22














            2












            2








            2





            $begingroup$

            Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.






            share|cite|improve this answer











            $endgroup$



            Expanding my comment into an answer: The term is homeomorphic. In particular, we call two topological spaces homeomorphic if there exists $f:Xto Y$ such that $f$ is a continuous bijection and $f^{-1}$ is continuous.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 7 '18 at 17:21

























            answered Dec 7 '18 at 16:52









            DreamConspiracyDreamConspiracy

            9391216




            9391216








            • 2




              $begingroup$
              Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
              $endgroup$
              – Noah Schweber
              Dec 7 '18 at 17:19












            • $begingroup$
              @NoahSchweber brain fart, thank you
              $endgroup$
              – DreamConspiracy
              Dec 7 '18 at 17:22














            • 2




              $begingroup$
              Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
              $endgroup$
              – Noah Schweber
              Dec 7 '18 at 17:19












            • $begingroup$
              @NoahSchweber brain fart, thank you
              $endgroup$
              – DreamConspiracy
              Dec 7 '18 at 17:22








            2




            2




            $begingroup$
            Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
            $endgroup$
            – Noah Schweber
            Dec 7 '18 at 17:19






            $begingroup$
            Note that the torus and Klein bottle aren't actually homeomorphic. (Also, the usual notation for the inverse is "$f^{-1}$.")
            $endgroup$
            – Noah Schweber
            Dec 7 '18 at 17:19














            $begingroup$
            @NoahSchweber brain fart, thank you
            $endgroup$
            – DreamConspiracy
            Dec 7 '18 at 17:22




            $begingroup$
            @NoahSchweber brain fart, thank you
            $endgroup$
            – DreamConspiracy
            Dec 7 '18 at 17:22


















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