Core spaces, sector spaces and fan spaces: a topological approach to domain theory

TL;DR

This paper explores the topological structures of core spaces, sector spaces, and fan spaces, revealing their interrelations and implications for domain theory, including characterizations of continuous domains and extensions of fundamental theorems.

Contribution

It introduces new characterizations of core and fan spaces, establishes categorical isomorphisms, and extends key theorems in domain theory using a topological approach.

Findings

Core spaces are characterized by supercompact neighborhood bases.

Sector spaces are obtained by joining topologies with dual specialization orders.

Continuous domains correspond to sober core spaces with Scott topology.

Abstract

We present old and new characterizations of core spaces, alias worldwide web spaces, originally defined by the existence of supercompact neighborhood bases. The patch spaces of core spaces, obtained by joining the original topology with a second topology having the dual specialization order, are the so-called sector spaces, which have good convexity and separation properties and determine the original space. The category of core spaces is shown to be concretely isomorphic to the category of fan spaces; these are certain quasi-ordered spaces having neighborhood bases of so-called fans, obtained by deleting a finite number of principal filters from a principal filter. This approach has useful consequences for domain theory. In fact, endowed with the Scott topology, the continuous domains are nothing but the sober core spaces, and endowed with the Lawson topology, they are the corresponding fan spaces. We generalize the characterization of continuous lattices as meet-continuous lattices with T$_2$ Lawson topology and extend the Fundamental Theorem of Compact Semilattices to non-complete structures. Finally, we investigate cardinal invariants like density and weight of the involved objects.