On A New Convergence Class in Sup-sober Spaces

Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of T0-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 andW. K. Ho by replacing directed subsets with irreducible sets. As a result, we formulate a new convergence class I in T0-spaces called Irr-convergence and establish that a sup-sober spaceX is SI-continuous if and only if it satisfies ∗-property and the convergence class I in it is topological.