# Can a simple closed curve in a compact surface be dense?

I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?

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 $$S^1$$.