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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!

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    $\begingroup$ You would need the Jordan curve theorem. $\endgroup$ – user98602 Dec 15 '18 at 15:42
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    $\begingroup$ @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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