On A New Convergence Class in Sup-sober Spaces

09/11/2017

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by   Hadrian Andradi, et al.

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