How can I prove that there is no closed smooth space filling curve from [0,1] to 2-dimensional sphere. I can construct a space filling curve to a cube in R^3, but why I can not expand it to sphere?
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You may construct such a curve over every compact space. – Blumer Nov 11 '17 at 11:49
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@Blumer But it will basically never be smooth. – Daniel Robert-Nicoud Nov 11 '17 at 12:02
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I didn’t understood the smoothness hypothesis. Sorry – Blumer Nov 11 '17 at 12:03
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As Daniel Robert-Nicoud hints in his reply to Blumer, "smooth" is the key word here. You can construct space-filling curves into the cube and the sphere, but neither one of those is smooth (i.e., possessing a continuous derivative). Proving that smooth curves cannot be space-filling requires some sophisticated analysis, though. – Paul Sinclair Nov 11 '17 at 16:21