from above and $x'^i_1 \to q$ from above.

If there is a sequence $\{x^i_0, x^i_1\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{X}}$ with $x^i_0 \to p$ from below and $x^i_1 \to q$ from above with $p \neq q$, then there is a sequence $\{x'^i_0, x'^i_1\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{X}}$ with $x'^i_0 \to p$ from above and $x'^i_1 \to q$ from above.

The following combinatorial notion was introduced in [@Ng def. 0.3].

\[The Ptolemy condition\] \[def:Ptolemy\] Let ${\mathscr{X}}$ be a set of diagonals of ${\mathscr{Z}}$. We say that [*${\mathscr{X}}$ satisfies the Ptolemy condition*]{} if, whenever $\{x_0,x_1\} \in {\mathscr{X}}$ and $\{y_0,y_1\} \in {\mathscr{X}}$ cross, then those of $\{x_0,y_0\}$, $\{x_0,y_1\}$, $\{x_1,y_0\}$ and $\{x_1,y_1\}$ which are diagonals of ${\mathscr{Z}}$ (i.e. whose vertices are non-neighbouring) also lie in ${\mathscr{X}}$. See Figure \[fig:Ptolemy\].

(0,0) circle(0.5cm);

(50:0.48) – (50:0.52); at (50:0.6) [$x_0$]{}; (110:0.48) – (110:0.52); at (110:0.6) [$y_0$]{}; (210:0.48) – (210:0.52); at (210:0.6) [$x_1$]{}; (300:0.48) – (300:0.52); at (300:0.6) [$y_1$]{};

(50:0.5) .. controls (50:0.2) and (210:0.2) .. (210:0.5); (110:0.5) .. controls (110:0.2) and (300:0.2) .. (300:0.5);

(50:0.5) .. controls (50:0.25) and (110:0.25) .. (110:0.5); (50:0.5) .. controls (50:0.2) and (300:0.2) .. (300:0.5); (210:0.5) .. controls (210:0.2) and (110:0.2) .. (110:0.5); (210:0.5) .. controls (210:0.2) and (300:0.2) .. (300:0.5);

