Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite.
E.Klarreich (see also U.Hamenstadt) proved the Gromov boundary of the curve complex of $S$ is bijective to the 
collection of ending laminations. Denote the collection of ending laminations by $B$.


Note: ending lamination implies that its complement in $S$ is a collection of (once-punctured) ideal polygons. 


My question is:
Given a number $N$, is there possible that there is a collection of essential simple closed curves $Y=\{c\_{\eta} \mid \eta\in B\}$ such that $d\_{\mathcal {C}(S)}(c\_{\eta}, c\_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta \neq \zeta$?



faraway curves in surface 
Are you trying to index a countable collection of curves by the (uncountable!) space of ending laminations? 


