We present an improved method for topology optimization with both adaptive mesh refinement and derefinement. Since the total volume fraction in topology optimization is usually modest, after a few initial iterations the domain of computation is largely void. Hence, it is inefficient to have many small elements, in such regions, that contribute significantly to the overall computational cost but contribute little to the accuracy of computation and design. At the same time, we want high spatial resolution for accurate three-dimensional designs to avoid postprocessing or interpretation as much as possible. Dynamic adaptive mesh refinement (AMR) offers the possibility to balance these two requirements. We discuss requirements on AMR for topology optimization and the algorithmic features to implement them. The numerical design problems demonstrate (1) that our AMR strategy for topology optimization leads to designs that are equivalent to optimal designs on uniform meshes, (2) how AMR strategies that do not satisfy the postulated requirements may lead to suboptimal designs, and (3) that our AMR strategy significantly reduces the time to compute optimal designs. ; Comment: adaptive mesh refinement, topology optimization, iterative solvers