On A New Convergence Class in Sup-sober Spaces

TL;DR

This paper extends domain theory to $T_0$-spaces by introducing a new convergence class called ${ ext{Irr}}$-convergence, characterizing when such spaces are ${ ext{SI}}^{-}$-continuous.

Contribution

It develops a topological parallel to a 2005 order-theoretic result by defining ${ ext{Irr}}$-convergence in $T_0$-spaces and characterizing ${ ext{SI}}^{-}$-continuity.

Findings

Introduces ${ ext{Irr}}$-convergence in $T_0$-spaces.

Characterizes ${ ext{SI}}^{-}$-continuous spaces via $*$-property and topological ${ ext{Irr}}$-convergence.

Provides a topological framework extending domain theory beyond posets.

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 Scott-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 $\mathcal{I}$ in $T_0$-spaces called ${\operatorname{Irr}}$-convergence and establish that a sup-sober space $X$ is ${\operatorname{SI}}^{-}$-continuous if and only if it satisfies $*$-property and the convergence class $\mathcal{I}$ in it is topological.