Topological Scott Convergence Theorem

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 $T_0$ 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., an order-theoretic characterisation of those posets for which the Scott-convergence $\mathcal{S}$ is topological. To do this, we make use of the $\mathcal{ID}$ replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., $\mathcal{I}$-continuous spaces correspond to continuous posets, as $\mathcal{I}$-convergence corresponds to $\mathcal{S}$-convergence. In this paper, we consider two novel topological concepts, namely, the $\mathcal{I}$-stable spaces and the $\mathcal{DI}$ spaces, and as a result we obtain some necessary (respectively, sufficient) conditions under which the convergence structure $\mathcal{I}$ is topological.