6

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?

enter image description here

João Alves Jr.
  • 837
  • 3
    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. – Adam Latosiński Jun 14 '19 at 20:21
  • 3
    From Wikipedia: There exist non-self-intersecting curves of nonzero area, the Osgood curves, but they are not space-filling. – Theo Bendit Jun 14 '19 at 20:23
  • 2
    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. – Robert Israel Jun 14 '19 at 20:25

3 Answers3

2

What is happening in the pictures that you drew is that you have a sequence of parameterized Jordan curves, i.e. continuous injective maps $f_n: S^1\to R^2$, and this sequence converges uniformly to a continuous map. However, a uniform limit of injective continuous maps need not be injective. The correct statement is:

Proposition. Suppose that $f: [0,1]\to R^2$ is a continuous injective map. Then the image of $f$ has empty interior. In particular, there cannot be a simple space filling curve or arc.

Proof. First of all, since $[0,1]$ is compact and $R^2$ is Hausdorff, $f: [0,1]\to E=f([0,1])$ is a homeomorphism.

Suppose that there exists an interior point $p\in E$. The set $E$ is path-connected. For every path-connected subset $A\subset R^2$ and every interior point $a\in A$, the complement $A\setminus \{a\}$ is path-connected. But $x=f^{-1}(p)$ disconnects $[0,1]$ unless $x$ is one of the end-points of $[0,1]$. Since the interior of $E$ contains infinitely many points, we can assume that $p$ is chosen so that $x\notin \{0,1\}$. Thus, $f^{-1}(E\setminus \{p\})$ is disconnected. It follows that $f: [0,1]\to E$ cannot be a homeomorphism. A contradiction. qed

Moishe Kohan
  • 97,719
0

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!

juan zaragoza
  • 141
  • This answer is wrong, Osgood does none of these. In fact, if $f: [0,1]\to R^2$ is a continuous injective map, then its image has empty interior. What Osgood constructs are Jordan curves of positive area. – Moishe Kohan Jun 28 '22 at 12:32
  • @MoisheKohan I am not sure how you have read Osgood's paper, but he does actually do "all of these". He does use, as you have said, the term Jordan curva: I find it missleading because, usually, Jordan curve refers to closed curves; that's why I use the term Jordan arc. The map is indeed injective, because every two different values $t_1,t_2\in [0,1]$ are mapped to different points in the square $[0,1]^2$. I would suggest you read Sagan's "Space-filling curves$ as well as this paper https://www.sciencedirect.com/science/article/pii/S0024379514007381?via%3Dihub Hope you finally understand it – juan zaragoza Jun 29 '22 at 13:45
  • It is not the matter of the end-points. Osgood constructs Jordan curves of positive area, now called Osgood curves. He does not construct bijective continuous maps $[0,1]\to [0,1]^2$ because such maps do not exist. – Moishe Kohan Jun 29 '22 at 15:26
  • I haven't said, at no point, that such curves are bijective. I have stated that they are INJECTIVE, becase they are. As a matter of fact, "osgood curves" is a term used for INJECTIVE curves, while the used term in general is "peano curve". Osgood's paper was interesting because he showed, for the first time, that there could be an injective space filling curve. I insist: you should read the paper cited above. – juan zaragoza Jun 29 '22 at 18:08
  • Space filling curves are required to be onto a square or a subset with nonempty interior. What is your definition of space-filling? – Moishe Kohan Jun 29 '22 at 18:37
  • Space filling curves do not need to be surjective. Usual definition is that a parametric curve $\gamma : [0,1]\longrightarrow \mathbb{R}^n$ is space-filling if $\gamma([0,1])$ has positive Jordan or Lebesgue measure. Osgood's curve is Lebesgue space-filling but not Jordan space-filling. This is all in the very first pages of the literature I refered you to. – juan zaragoza Jun 30 '22 at 09:36
  • I see: The authors of the paper and you are using a nonstandard terminology, likely not what OP had in mind. (In contrast, Osgood himself does not refer to his curve as "space-filling.") The standard definition of space-filling is in this Wikipedia article. Curves of positive area are usually called Osgood curves, not space-filling. My suggestion is to edit your answer to make it clear what notion of space-filling you are referring to. – Moishe Kohan Jun 30 '22 at 11:27
  • Wikipedia's definition is not the standard definition and I bet it is the only one you have read prior to this conversation. Sagan, in his book "Space-Filling curves" uses the same definition as the authors of the paper; Sagan's book is the main reference regarding this topic and considered as "the bible" os space-filling curves. Of course Osgood didn't talk about space-filling curves; Camille Jordan was a contemporary of his and Lebesgue didn't publish his thesis until one year after Osgood's paper was published. My suggestion is that you read more things than wikipedia before speaking. – juan zaragoza Jul 02 '22 at 15:32
  • You loose your bet. The same definition is used in numerous other books in analysis and topology. – Moishe Kohan Jul 02 '22 at 15:42
0

The Moore curve is closed in the limit. Start and end are neighboring in each iteration, and the distance between them decreases approximately as $2^{-n}$.

A. Donda
  • 1,458