 Filling set of curves

A collection of essential simple closed curves fills a surface if every essential simple closed curve has positive geometric intersection number with at least one member. For such curves in minimal position of curves or arcs on a closed surface of positive genus, this is equivalent to every complementary component being a disc.

The Alexander method uses a stronger cutting condition when boundary or marked points are present: an Alexander system must cut the surface into discs with at most one marked point each. For example, a pair of pants has no essential nonperipheral simple closed curves, so the empty collection passes the curve-intersection test, but it does not cut the surface into discs. The boundary-relative version must therefore keep the stronger condition explicit.