How to prove that a Jordan curve cannot be everywhere dense in $[0,1]^2$ ?
2026-04-07 00:46:02.1775522762
A Jordan curve cannot be everywhere dense in $[0,1]^2$
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Using Hellen’s comment for answer.
A Jordan curve's image is compact. Therefore the image would be the whole square. It is also a homeomorphism to $S^1$. Now, remove two points from $S^1$ and the corresponding points from the square, and compare connectedness of the results.