My question is about a simple closed curve that is also a space-filling curve. The figure shows 6 iterations of the formation of a Hilbert curve (limit), whose trace is a solid square. I think we may, at each iteration, connect the endpoints of the curve in order to obtain a Jordan curve (simple closed curve), preserving the limit of this sequence of curves (a solid square). So, at the limit, we will have a space-filling, simple closed curve. By the Jordan curve theorem, every simple closed curve "divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere" (Wikipedia).

Question: Does there really exist a space-filling, simple closed curve? What is the interior region of a space-filling, simple closed curve? The empty set?

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    $\begingroup$ I don't think the curve that you get in the limit is still simple. It seems to me that some, if not all, points of the square are (in the limit) covered several times. $\endgroup$ Jun 14 '19 at 20:21
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    $\begingroup$ From Wikipedia: There exist non-self-intersecting curves of nonzero area, the Osgood curves, but they are not space-filling. $\endgroup$ Jun 14 '19 at 20:23
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    $\begingroup$ Of course its not simple: a simple curve is the image of a closed interval (or circle, if it's a closed curve) under a continuous one-to-one map. A continuous one-to-one function from a compact set to Hausdorff space is a homeomorphism. A (filled) square is not homeomorphic to an interval or circle. $\endgroup$ Jun 14 '19 at 20:25

In Osgood's paper "A Jordan Curve of Positive Area" you have the PDF here he provides a construction for a space-filling curve $[0,1]\hookrightarrow [0,1]^2$ but it is not a closed curve: it is, using nomenclature of the Jordan Curve Theorem, a Jordan Arc. Still, at the end of the paper, he provides the construction of a closed jordan curve. Hope it satisfies your curiosity!


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