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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?

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A simple closed curve in a surface $X$ is a continuous injection $f:S^1\to X$. 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 $S^1$.

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  • $\begingroup$ Aha! I see... stupid question... Thanks! $\endgroup$ – chikurin Jan 27 at 4:10

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