Filling set of curves

A collection of curves fills a surface when every essential simple closed curve intersects at least one member. For curves in minimal position on a closed surface, this is equivalent to every complementary component being a disc.

Split a closed genus-two surface along a separating curve $\alpha$ into two one-holed tori. In each torus choose two disjoint proper arcs that cut it into a disc, and match their four endpoints across $\alpha$ so that the four arcs join into one simple closed curve $\beta$. Then $i(\alpha ,\beta )=4$ and the complement of $\alpha \cup \beta$ is two discs, so the pair fills.