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
asked Dec 15 '18 at 14:44
JiuJiu
515113
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add a comment |
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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
asked Dec 15 '18 at 14:44
JiuJiu
515113
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1
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You would need the Jordan curve theorem.
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– Mike Miller
Dec 15 '18 at 15:42
1
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@MikeMiller thanks for your comment. J’adore l’invitation au voyage avec la musique par Duparc :D
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– Jiu
Dec 15 '18 at 16:06
add a comment |
$begingroup$
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
asked Dec 15 '18 at 14:44
JiuJiu
515113
$endgroup$
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
asked Dec 15 '18 at 14:44
JiuJiu
515113
asked Dec 15 '18 at 14:44
JiuJiu
515113
asked Dec 15 '18 at 14:44
JiuJiu
515113
asked Dec 15 '18 at 14:44
JiuJiu
515113
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
add a comment |
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
add a comment |
1 Answer
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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.
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
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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.
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
$endgroup$
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.
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
answered Dec 15 '18 at 17:10
Paul FrostPaul Frost
10.6k3933
10.6k3933
add a comment |
add a comment |
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You would need the Jordan curve theorem.
$endgroup$
– Mike Miller
Dec 15 '18 at 15:42
1
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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