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!
2026-05-05 21:27:42.1778016462
Prove that sphere and torus are not homeomorphic using general topology only?
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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.