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Zbl 1087.06004
Zhao, Bin; Zhao, Dongsheng
Lim-inf convergence in partially ordered sets.
(English)
[J] J. Math. Anal. Appl. 309, No. 2, 701-708 (2005). ISSN 0022-247X

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.
[J. D. Lawson (Baton Rouge)]
MSC 2000:
*06B35 Continuous lattices

Keywords: lim-inf-convergence; continuous posets; lim-inf$_2$-convergence; $\alpha$-continuous posets; Scott topology

Citations: Zbl 1088.06001

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