  In the curve complex for a surface, a handlebody set is the set of loops that
bound properly embedded disks in a given handlebody bounded by the surface. A
boundary set is the set of non-separating loops in the curve complex that bound
two-sided, properly embedded surfaces. For a Heegaard splitting, the distance
between the boundary sets of the handlebodies is zero if and only if the
ambient manifold contains a non-separating, two sided incompressible surface.
We show that the boundary set is 2-dense in the curve complex, i.e. every
vertex is within two edges of a point in the boundary set.
