Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that there is at least one intersection point between the geodesic representative of $\alpha$ and the geodesic representative of $\beta$ which is not a double point.
Q) Is $X$ nowhere dense in $\mathcal{T}(S)$?
I think it is true but I can't prove it.