In this paper, we define and study irreducible convergence and irreducible order-convergence in spaces, which can be seen as topological counterparts of lim-inf-convergence and order-convergence in posets, respectively. Especially, we give sufficient and necessary conditions for irreducible convergence and irreducible order-convergence in spaces to be topological.
"Irreducible convergence in $T_0$ spaces." Rocky Mountain J. Math. 50 (1) 337 - 353, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.337