Irreducible convergence in $T_0$ spaces

Abstract

In this paper, we define and study irreducible convergence and irreducible order-convergence in $T_0$ 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 $T_0$ spaces to be topological.