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Space-filling curves, constructively
filling curves, constructively - 30 January 2024 - Constructive math, Gems and stones In 1890 Giuseppe Peano discovered a square-filling curve, and a year later David Hilbert published his variation. In those days people did not waste readers' attention with dribble – Peano explained it all on 3 pages, and Hilbert on just 2 pages, with a picture! But are these constructive square-filling curves? There's no doubt that the curves themselves are defined constructively, for instance as limits of uniformly continuous maps.
Recall that $\gamma$ is self-similar, as it is made of four copies of itself, each scaled by a factor $1/2$, translated and rotated to cover precisely one quarter of the unit square. To summarize the argument: in a topos of sheaves “$\gamma$ is surjective” is a very strong condition, namely that $\gamma$ has local sections – and these may not exist for topological or geometric reasons. At the same time we still have Theorem 1, so also in a topos of sheaves the usual Hilbert curve leaves no empty space in the unit square.
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