Punctured three-sphere

A punctured three-sphere is $S^3$ with the interiors of finitely many disjoint piecewise linear balls removed. Capping its spherical boundary components with balls restores $S^3$. Removing another ball or gluing two such manifolds along one spherical boundary component preserves this class.

Suppose disjoint separating spheres cut a 3-manifold into pieces none of which is a punctured three-sphere. Compress one sphere along a disk disjoint from the others. The old sphere and the two new spheres enclose a three-times punctured sphere. At least one of the two pieces on the compressed side is not a punctured sphere; retain that new sphere and discard the other. The piece on its opposite side cannot become a punctured sphere, since capping its boundaries and applying the Three-dimensional Schoenflies theorem would make the original opposite piece a punctured sphere as well. Hence the number of spheres and the nontriviality condition can be preserved.