Topological Scott Convergence Theorem

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., 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 T0 spaces) we are working in. To tackle each of these difficulties, we introduce a new type of T0 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.