= Filling set of curves
= Fill a surface
{synonym}
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 \b[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 (mathematics)> 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.
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