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Can a simple closed curve in a compact surface be dense?
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I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
general-topology geometric-topology
asked Jan 27 at 4:00
chikurinchikurin
1109
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add a comment |
$begingroup$
I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
general-topology geometric-topology
asked Jan 27 at 4:00
chikurinchikurin
1109
$endgroup$
add a comment |
$begingroup$
I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
general-topology geometric-topology
asked Jan 27 at 4:00
chikurinchikurin
1109
$endgroup$
I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
general-topology geometric-topology
general-topology geometric-topology
asked Jan 27 at 4:00
chikurinchikurin
1109
asked Jan 27 at 4:00
chikurinchikurin
1109
asked Jan 27 at 4:00
chikurinchikurin
1109
asked Jan 27 at 4:00
chikurinchikurin
1109
asked Jan 27 at 4:00
chikurinchikurin
1109
1109
add a comment |
add a comment |
1 Answer
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$begingroup$
A simple closed curve in a surface $X$ is a continuous injection $f:S^1to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
$endgroup$
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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$begingroup$
A simple closed curve in a surface $X$ is a continuous injection $f:S^1to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
$endgroup$
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
add a comment |
$begingroup$
A simple closed curve in a surface $X$ is a continuous injection $f:S^1to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
$endgroup$
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
add a comment |
$begingroup$
A simple closed curve in a surface $X$ is a continuous injection $f:S^1to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
$endgroup$
A simple closed curve in a surface $X$ is a continuous injection $f:S^1to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
answered Jan 27 at 4:07
Eric WofseyEric Wofsey
190k14216348
190k14216348
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
add a comment |
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
$begingroup$
Aha! I see... stupid question... Thanks!
$endgroup$
– chikurin
Jan 27 at 4:10
add a comment |
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