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Is there a Hilbert curve for every space? as I understanding it in layman terms, it's a way to order a space sequentially which visits every place once.

Does it work with continuous spaces, or just discrete? Are there certain limitations of dimensionality that you cannot use a Hilbert curve to explore?

I guess I'm asking if everything can be serialized?

What must I understand to understand the boundary of the Hilbert Curve concept?

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    I'm pretty sure it was well known just after Peano's 1890 paper appeared that extensions to higher dimensions can be made, as Peano mentions a continuous curve filling a $3$-dimensional cube at the end of his 1890 paper. The first published explicit construction I know of is: Jean-Armand Marie Joseph de Séguier (1862-1935), Courbe remplissant un cube à $n$ dimensions [Curve filling a cube of $n$ dimensions], Bulletin de la Société Mathématique de France 29 (1901), 312-314. (continued) – Dave L. Renfro Jul 05 '23 at 15:51
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    I believe what de Séguier did was to carry out the details for a generalization of Hilbert's 1891 approach. For further generalizations, see the Hahn-Mazurkiewicz theorem. – Dave L. Renfro Jul 05 '23 at 15:51
  • A Hilbert curve (a continuous surjective map from $[0,1] \to [0,1]^2$) does not "visit every place once". Many, many points are visited multiple times - some of them may even be visited infinitely many times. While continuous injective maps, continuous surjective maps, and discontinuous bijective maps all exist, there is no continuous bijective map between $[0,1]$ and $[0,1]^2$. – Paul Sinclair Jul 06 '23 at 14:10

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