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Lim-inf convergence in partially ordered sets. (English) Zbl 1087.06004
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 {\it G. Gierz}, {\it K. Hofmann}, {\it K. Keimel}, {\it J. Lawson}, {\it M. Mislove} and {\it 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.

06B35Continuous lattices and posets, applications
Full Text: DOI
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