New concepts of limes inferior and limes superior in the space of von Neumann algebras on a fixed Hilbert space are defined, and the topology corresponding to these notions is related to earlier works of Effros and Marechal. A main technical result is that the commutant operation is a homeomorphism on the space of von Neumann algebras with this topology. Further, the topological properties of several classes and types of von Neumann factors (regarded as subspaces) are determined, and also continuity-type results for Tomita-Takesaki theory are proved. Some applications to subfactor theory are given.