Testwiki:Reference desk/Archives/Mathematics/2023 October 9

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October 9

Name of a theorem

I vaguely remember there being a theorem listed in some Wikipedia article that states that, given a countable set of points $P$ in Euclidean space ${ℝ}^n$, one can construct a continuous function $C:ℝ\to {ℝ}^n$ that passes through all such points (i.e. $P\subset Im(C)$.) I may have the details of the theorem wrong, perhaps $P$ is bounded and $C$ is actually on the domain $[0,1]$. In any case, I also remember that this statement accompanied a picture of some fractal-like set of points, with a description of the picture stating that a consequence of the theorem is that there is a curve that passes through all points in the set. I've had no luck finding the name of this theorem, and was wondering if anyone might happen to remember what it is? GalacticShoe (talk) 15:01, 9 October 2023 (UTC)

I never heard of this theorem, but it is almost trivial to construct such a function. Let $(p_i{)}_{i\in \mathbb{N}}$ be an enumeration of $P,$ and let $(x_{i1},x_{i2},...,x_{in})=p_i.$ So $x_{ij}={\pi }_j(p_i),$ where ${\pi }_j:{ℝ}^n\to ℝ$ selects the $j$th coordinate of its argument. For each $j\in \{1,...,n\},$ construct a continuous function $c_j:ℝ\to ℝ$ such that $c_j(i)=x_{ij}$ for all $i\in \mathbb{N}.$ Then define $C$ to be the unique function such that ${\pi }_j\circ C=c_j,j\in \{1,...,n\}.$  --Lambiam 15:56, 9 October 2023 (UTC)

As it turns out, the reason why the theorem I came up with was trivial was because I completely misremembered it. I was thinking of the Denjoy–Riesz theorem, where $P$, rather being countable, is compact and totally disconnected, and where the continuous $C$, in addition to being on $[0,1]$, is also simple. Chances are I came across this while looking at the page on the Jordan curve theorem, then hid it away in my memory. GalacticShoe (talk) 16:09, 9 October 2023 (UTC)

If $P$ is not countable but bounded, iterate the construction of (for example) the Hilbert curve to get an $n$-dimensional hypercube-filling curve, and scale it up so that it envelops $P.$  --Lambiam 16:09, 9 October 2023 (UTC)