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 nonseparating
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 nonseparating, two sided
incompressible surface. We show that every vertex in the curve complex is within two
edges of a point in the boundary set.