# Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.

Let $$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$ for all $n\in\mathbb{N}$.

Then since $C_{n}$ is a finite subset of $[0,1]^{2}$, one can talk about the convergence of $C_{n}$ in the Hausdorff metric (i.e. the convergence in the hyperspace topology or the Vietoris topology).

Does $(C_{n})_{n\in\mathbb{N}}$ converge in the Hausdorff metric to $S\cup L$ where $S$ is the Sierpinski triangle in the upper-left half of $[0,1]\times[0,1]$ and $L=[0,1]\times\{1\}$?

Computer calculations seem to suggest that $C_{n}\rightarrow S\cup L$.

• On this page boolesrings.org/jvanname/…, we can reorder the elements of $A_{n}$ in a canonical way so that the compact set $B_{n}$ corresponding to $C_{n}$ (composed of squares instead of points) are strictly decreasing and therefore converge to a compact set $B$. However, the compact $B$ is a proper subset of a Sierpinski-like fractal. – Joseph Van Name Dec 1 '16 at 15:54