Topological Scott Convergence Theorem
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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 with an attempt to formulate a topological version of the Scott Convergence Theorem, i.e., order-theoretic characterisation of those posets for which the Scott-convergence, S, is topological. To do this, we make use of the so-called Zhao-Ho's replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., Irr-continuous space is to continuous poset, as I-convergence is to S-convergence. In this paper, we want to highlight the difficulties involved in our attempt because of the much more general ambient environment (of T_0 spaces) we are working in. To tackle each of these difficulties, we introduce a new type of T_0 spaces to overcome it. In this paper, we consider two novel topological concepts, namely, the balanced spaces and the nice spaces, and as a result we obtain some necessary (respectively, sufficient) condition for which the new convergence structure, I, is topological.
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