A New Convergence Inducing the SI-topology

Abstract

In their attempt to develop domain theory in situ T0 spaces, Zhao and Ho introduced a new topology defined by irreducible sets of a resident topological space, called the SI-topology. Notably, the SI-topology of the Alexandroff topology of posets is exactly the Scott topology, and so the SI-topology can be seen as a generalisation of the Scott topology in the context of general T0 spaces. It is well known that the convergence class that induces the Scott topology is the Scott-convergence – also known as lim-inf convergence by some authors. Till now, it is not known which convergence structure induces the SI- topology of a given T0 space. In this paper, we fill in this gap in the literature by providing a convergence structure, called the SI-convergence structure, that induces the SI-topology. Additionally, we introduce the notion of I-continuity that is closely related to the SI-convergence structure, but distinct from the existing notion of SI-continuity (invented by Zhao and Ho earlier). For SI-continuity, we obtain here a somewhat surprising equivalent condition for it. Finally, we give some examples of non-Alexandroff SI-continuous spaces.