Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 4 / b / Solution

Curves ${\alpha }_1,\dots ,{\alpha }_n$ in minimal position fill a surface $S$ when every essential simple closed curve has positive geometric intersection with at least one ${\alpha }_i$. On a closed surface this is equivalent to every component ofbeing a disc: a non-disc complementary component contains an essential curve, while any curve disjoint from the collection lies in such a component.