  Different notions for order convergence have been considered by various
authors. Associated to every notion of order convergence corresponds a
topology, defined by taking as the closed sets those subsets of the poset
satisfying that no net in them order converges to a point that is outside of
the set. We shall give a thorough overview of these different notions and
provide a systematic comparison of the associated topologies. Then, in the last
section we shall give an application of this study by giving a result on von
Neumann algebras complementing the study started in \cite{ChHaWe}. We show that
for every atomic von Neumann algebra (not necessarily $\sigma$-finite) the
restriction of the order topology to bounded parts of $M$ coincides with the
restriction of the $\sigma$-strong topology $s(M,M_\ast)$. We recall that the
methods of \cite{ChHaWe} rest heavily on the assumption of $\sigma$-finiteness.
Further to this, for a semi-finite measure space, we shall give a complete
picture of the relations between the topologies on $L^\infty$ associated with
the duality $\langle L^1, L^\infty\rangle$ and its order topology.
