In their attempt to develop domain theory in situ T 0 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 T 0 spaces. It is well known that the convergence structure 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 T 0 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 (introduced by Zhao and Ho earlier). For SI-continuity, we obtain here some equivalent conditions for it. Finally, we give some examples of non-Alexandroff SI-continuous 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 with an attempt to formulate a topological version of the Scott Convergence Theorem, i.e., an order-theoretic characterisation of those posets for which the Scott-convergence S is topological. To do this, we make use of the ID replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., I-continuous spaces correspond to continuous posets, as I-convergence corresponds to S-convergence. In this paper, we consider two novel topological concepts, namely, the I-stable spaces and the DI spaces, and as a result we obtain some necessary (respectively, sufficient) conditions under which the convergence structure I is topological.Theorem 1.1 (Scott convergence theorem [GHK + 03, Theorem II-1.9]). For a directed complete poset P , the following are equivalent:
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 and W. 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 space X is SI − -continuous if and only if it satisfies * -property and the convergence class I in it is topological.
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 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 T0 spaces called Irr-convergence and established that this convergence class I on a k-bounded sober space X is topological if and only if X is Irr-continuous.
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