On a new convergence class in k-bounded sober spaces

TL;DR

This paper develops a new topological convergence class called Irr-convergence in $T_0$ spaces, extending domain theory beyond posets, and characterizes when this convergence is topological in $k$-bounded sober spaces.

Contribution

It introduces Irr-convergence as a topological class in $T_0$ spaces and characterizes its topological nature in $k$-bounded sober spaces, extending domain theory.

Findings

Irr-convergence is topological iff the space is Irr-continuous.

Extension of domain theory to $T_0$ spaces using irreducible sets.

Provides a topological parallel to a known order-theoretic result.

Abstract

Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of $T_0$ spaces instead of restricting to posets. In this paper, we respond to this calling by proving a topological parallel of a 2005 result due to B. Zhao and D. Zhao, i.e., an order-theoretic characterisation of those posets for which the lim-inf convergence is topological. We do this by adopting a recent approach due to D. Zhao and W. K. Ho by replacing directed subsets with irreducible sets. As a result, we formulate a new convergence class on $T_0$ spaces called Irr-convergence and established that this convergence class $\mathcal{I}$ on a $k$-bounded sober space $X$ is topological if and only if $X$ is Irr-continuous.