Topological Scott Convergence Theorem
Hadrian Andradi, Weng Kin Ho

TL;DR
This paper develops a topological version of the Scott Convergence Theorem by introducing new concepts like $\\ extit{\mathcal{I}}$-stable and $\mathcal{DI}$ spaces, linking order-theoretic and topological convergence.
Contribution
It formulates a topological analogue of the Scott Convergence Theorem using the $\mathcal{ID}$ replacement principle and introduces new topological concepts.
Findings
Identifies conditions for $\mathcal{I}$-convergence to be topological.
Establishes the relationship between $\mathcal{I}$-stable spaces and $\mathcal{DI}$ spaces.
Provides necessary and sufficient conditions for topological convergence structures.
Abstract
Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of 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 is topological. To do this, we make use of the replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., -continuous spaces correspond to continuous posets, as -convergence corresponds to -convergence. In this paper, we consider two novel topological concepts, namely, the -stable spaces and the spaces, and as a result we obtain some necessary…
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Taxonomy
TopicsConstraint Satisfaction and Optimization · Advanced Topology and Set Theory
