  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:
  The weak ideal property implies topological dimension zero.
  For a separable C*-algebra~A, topological dimension zero is equivalent to RR
(O_2 \otimes A) = 0, to D \otimes 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_{\infty} \otimes B contains a nonzero projection.
  Extending the known result for Z_2, the classes of C*-algebras with
topological dimension zero, with the weak ideal property, and with residual
(SP) are closed under crossed products by arbitrary actions of abelian
2-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 \otimes_{min} B has the weak ideal
property.
  If X is a totally disconnected locally compact Hausdorff space and A is a C_0
(X)-algebra all of whose fibers have one of the weak ideal property,
topological dimension zero, residual (SP), or the combination of pure
infiniteness 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.
  The 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.
