Using standard methods of replacing the infima of residual subsets of nets by directed sets with elements eventual lower bounds, the authors study a notion of lim-inf convergence of nets in general partially ordered sets. The main result derived is that the lim-inf convergence so defined is topological if and only if the poset in question is a continuous poset. It is not difficult to see, although the authors do not point it out, that the topology generated from their notion of lim-inf convergence is the Scott topology. Closely related results may be found in Chapter II-1 of Continuous lattices and domains by G. Gierz
, K. Hofmann
, K. Keimel
, J. Lawson
, M. Mislove
and D. S. Scott
[Cambridge University Press, Cambridge (2003; Zbl 1088.06001
)]. The authors also introduce a weaker form of lim-inf convergence and a corresponding notion of continuity of a poset and again show that convergence is topological if and only if the poset is continuous in this alternative sense.