Abstract. Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include:• e weak ideal property implies topological dimension zero.• For a separable C*-algebra A, topological dimension zero is equivalent to RR(O ⊗ A) = , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C*-algebras B such that O ∞ ⊗ B contains a nonzero projection.• Extending the known result for Z , the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) in nite, or which are purely in nite and have the ideal property, are closed under crossed products by arbitrary actions of abelian -groups.• If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ min B has the weak ideal property.• If X is a totally disconnected locally compact Hausdor space and A is a C (X)-algebra all of whose bers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure in niteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).• Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras.• e weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.