we expect cluster tilting subcategories of $\cC( \cZ )$ to correspond to triangulations of the $\infty$-gon with vertex set $\cZ$.

This is, in a sense, true, but there is more to say: The definition of admissible subset permits $\cZ$ to have a complicated configuration of limit points, and it is crucial how the endpoints of diagonals converge to the limit points. Hence the following two definitions.

\[The proper limit points of $\cZ$\] We denote by $\overline{{\mathscr{Z}}}$ the topological closure of ${\mathscr{Z}}$ in $S^1$, and by $$L({\mathscr{Z}}) = \overline{{\mathscr{Z}}} \setminus {\mathscr{Z}}$$ the set of proper limit points of $\cZ$. It is disjoint from $\cZ$ because $\cZ$ is discrete.

\[Leapfrogs and fountains\] \[def:leapfrog\_and\_fountain\] Let ${\mathscr{X}}$ be a set of diagonals of ${\mathscr{Z}}$. The following notions are illustrated by Figure \[fig:cluster tilting subcategory\].

(70:0.5) .. controls (70:0.3) and (120:0.3) .. (120:0.5); (70:0.5) .. controls (70:0.3) and (115:0.3) .. (115:0.5); (70:0.5) .. controls (70:0.3) and (110:0.3) .. (110:0.5); (70:0.5) .. controls (70:0.3) and (105:0.3) .. (105:0.5);

(70:0.5) .. controls (70:0.2) and (230:0.2) .. (230:0.5); (70:0.5) .. controls (70:0.2) and (235:0.2) .. (235:0.5); (70:0.5) .. controls (70:0.2) and (240:0.2) .. (240:0.5); (70:0.5) .. controls (70:0.2) and (245:0.2) .. (245:0.5);

(215:0.4) arc (215:225:0.4);

(70:0.5) .. controls 