Web spaces and worldwide web spaces: topological aspects of domain theory

TL;DR

This paper explores topological generalizations of domain theory called web spaces and C-spaces, providing new characterizations, categorical isomorphisms, and implications for continuous lattices and cardinal invariants.

Contribution

It introduces new characterizations of web spaces and C-spaces, establishes their categorical equivalence with fan spaces, and extends domain-theoretic results to non-complete posets.

Findings

Web spaces and C-spaces are characterized via patch spaces with dual topologies.

Category of C-spaces is isomorphic to fan spaces, with specific neighborhood bases.

Generalized Scott spaces correspond to locally and globally approximating ideal extensions.

Abstract

Web spaces, wide web spaces and worldwide web spaces (alias C-spaces) provide useful generalizations of continuous domains. We present new characterizations of such spaces and their patch spaces, obtained by joining the original topology with a second topology having the dual specialization order; these patch spaces possess good convexity and separation properties and determine the original web spaces. The category of C-spaces is concretely isomorphic to the category of fan spaces; these are certain quasi-ordered spaces having neighborhood bases of fans, obtained by deleting a finite number of principal dual ideals from a principal dual ideal. Our approach has useful consequences for domain theory, because the T$_0$ web spaces are exactly the generalized Scott spaces of locally approximating ideal extensions, and the T$_0$ C-spaces are exactly the generalized Scott spaces of globally approximating and interpolating ideal extensions. We extend the characterization of continuous lattices as meet-continuous lattices with T$_2$ Lawson topology and the Fundamental Theorem of Compact Semilattices to non-complete posets. Finally, cardinal invariants like density and weight of the involved objects are investigated.