We give a thorough overview of the different notions for order convergence that are found in the literature and provide a systematic comparison of the associated topologies. As an application of this study we prove a result related to the order topology on von Neumann algebras, complementing the study started in Chetcuti et al. (Stud. Math. 230:95-120, 2015). We show that for every atomic von Neumann algebra (not necessarily σ-finite) the restriction of the order topology to bounded parts of M coincides with the restriction of the σ-strong topology s(M, M∗). We recall that the methods of Chetcuti et al. (Stud. Math. 230:95-120, 2015) rest heavily on the assumption of σ-finiteness. Furthermore, for a semi-finite measure space, we provide a complete picture of the relations between the topologies on L∞ associated with the duality ⟨ L1, L∞⟩ and its order topology.
