Prove that sphere and torus are not homeomorphic using general topology only?












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Is there a way of proving that the sphere and the torus are not homeomorphic without the tools of algebraic topology? For example I want to say that you can remove a circle from a torus and it is still connected, but if you remove a circle from a sphere then it is no longer connected. But I don’t know how to prove that the homeomorphic image of a circle from the torus is still something like a circle on the sphere so that you can speak about the interior and exterior of it . Can you help? Thanks in advance!






general-topology







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    You would need the Jordan curve theorem.
    $endgroup$
    – Mike Miller
    Dec 15 '18 at 15:42






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    @MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
    $endgroup$
    – Jiu
    Dec 15 '18 at 16:06
















2












$begingroup$


Is there a way of proving that the sphere and the torus are not homeomorphic without the tools of algebraic topology? For example I want to say that you can remove a circle from a torus and it is still connected, but if you remove a circle from a sphere then it is no longer connected. But I don’t know how to prove that the homeomorphic image of a circle from the torus is still something like a circle on the sphere so that you can speak about the interior and exterior of it . Can you help? Thanks in advance!






general-topology







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    You would need the Jordan curve theorem.
    $endgroup$
    – Mike Miller
    Dec 15 '18 at 15:42






  • 1




    $begingroup$
    @MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
    $endgroup$
    – Jiu
    Dec 15 '18 at 16:06














2












2








2


0



$begingroup$


Is there a way of proving that the sphere and the torus are not homeomorphic without the tools of algebraic topology? For example I want to say that you can remove a circle from a torus and it is still connected, but if you remove a circle from a sphere then it is no longer connected. But I don’t know how to prove that the homeomorphic image of a circle from the torus is still something like a circle on the sphere so that you can speak about the interior and exterior of it . Can you help? Thanks in advance!






general-topology







share|cite|improve this question









$endgroup$




Is there a way of proving that the sphere and the torus are not homeomorphic without the tools of algebraic topology? For example I want to say that you can remove a circle from a torus and it is still connected, but if you remove a circle from a sphere then it is no longer connected. But I don’t know how to prove that the homeomorphic image of a circle from the torus is still something like a circle on the sphere so that you can speak about the interior and exterior of it . Can you help? Thanks in advance!






general-topology




general-topology






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asked Dec 15 '18 at 14:44









JiuJiu

515113




515113








  • 1




    $begingroup$
    You would need the Jordan curve theorem.
    $endgroup$
    – Mike Miller
    Dec 15 '18 at 15:42






  • 1




    $begingroup$
    @MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
    $endgroup$
    – Jiu
    Dec 15 '18 at 16:06














  • 1




    $begingroup$
    You would need the Jordan curve theorem.
    $endgroup$
    – Mike Miller
    Dec 15 '18 at 15:42






  • 1




    $begingroup$
    @MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
    $endgroup$
    – Jiu
    Dec 15 '18 at 16:06








1




1




$begingroup$
You would need the Jordan curve theorem.
$endgroup$
– Mike Miller
Dec 15 '18 at 15:42




$begingroup$
You would need the Jordan curve theorem.
$endgroup$
– Mike Miller
Dec 15 '18 at 15:42




1




1




$begingroup$
@MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
$endgroup$
– Jiu
Dec 15 '18 at 16:06




$begingroup$
@MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
$endgroup$
– Jiu
Dec 15 '18 at 16:06










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As Mike Miller pointed out, you need the Jordan curve thorem to show that any homeomorphic image of a circle in a sphere separates the sphere in two components. I recommend to have a look at the history of the proof in https://en.wikipedia.org/wiki/Jordan_curve_theorem. The "early proofs" are not based on the machinery of algebraic topology. See for example



Veblen, Oswald. "Theory on plane curves in non-metrical analysis situs." Transactions of the American Mathematical Society 6.1 (1905): 83-98.






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    $begingroup$

    As Mike Miller pointed out, you need the Jordan curve thorem to show that any homeomorphic image of a circle in a sphere separates the sphere in two components. I recommend to have a look at the history of the proof in https://en.wikipedia.org/wiki/Jordan_curve_theorem. The "early proofs" are not based on the machinery of algebraic topology. See for example



    Veblen, Oswald. "Theory on plane curves in non-metrical analysis situs." Transactions of the American Mathematical Society 6.1 (1905): 83-98.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      As Mike Miller pointed out, you need the Jordan curve thorem to show that any homeomorphic image of a circle in a sphere separates the sphere in two components. I recommend to have a look at the history of the proof in https://en.wikipedia.org/wiki/Jordan_curve_theorem. The "early proofs" are not based on the machinery of algebraic topology. See for example



      Veblen, Oswald. "Theory on plane curves in non-metrical analysis situs." Transactions of the American Mathematical Society 6.1 (1905): 83-98.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        As Mike Miller pointed out, you need the Jordan curve thorem to show that any homeomorphic image of a circle in a sphere separates the sphere in two components. I recommend to have a look at the history of the proof in https://en.wikipedia.org/wiki/Jordan_curve_theorem. The "early proofs" are not based on the machinery of algebraic topology. See for example



        Veblen, Oswald. "Theory on plane curves in non-metrical analysis situs." Transactions of the American Mathematical Society 6.1 (1905): 83-98.






        share|cite|improve this answer









        $endgroup$



        As Mike Miller pointed out, you need the Jordan curve thorem to show that any homeomorphic image of a circle in a sphere separates the sphere in two components. I recommend to have a look at the history of the proof in https://en.wikipedia.org/wiki/Jordan_curve_theorem. The "early proofs" are not based on the machinery of algebraic topology. See for example



        Veblen, Oswald. "Theory on plane curves in non-metrical analysis situs." Transactions of the American Mathematical Society 6.1 (1905): 83-98.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 15 '18 at 17:10









        Paul FrostPaul Frost

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