A study of convergences in partially ordered sets (Q1985617)

In [Proc. Lond. Math. Soc., II. Ser. 52, 386--400 (1951; Zbl 0042.41004)], \textit{B. C. Rennie} introduced and studied \(o_2\)-convergence in posets. It is known that this convergence is not topological. In [Sci. Math. Jpn. 83, No. 1, 23--38 (2020; Zbl 1468.54005)], the authors of this paper introduced the ideal version of \(o_2\)-convergence of nets, called ideal-\(o_2\)-convergence, and proved a result stating that the ideal-\(o_2\)-convergence in a poset is topological if and only if the poset is \(O_2\)-doubly continuous, cf. \textit{Q. Li} and \textit{Z. Zou} [Open Math. 14, 205--211 (2016; Zbl 1348.54005)]). Here, the authors give an alternative proof of this result. The notion of ideal-lim-inf-convergence on a poset is introduced and it is proved that it is topological if and only if the poset is continuous. It is also proved that the ideal-lim-inf-topology, the lim-inf-topology and the Scott topology on a poset coincide.