Continuity and core compactness of topological spaces

TL;DR

This paper explores the properties of directed topological spaces, introducing new approximation relations that extend the way-below relation from domain theory, and establishes key characterizations of continuity and core compactness.

Contribution

It introduces n- and d-approximations for T0 spaces, extending domain-theoretic concepts to topological spaces and characterizing various forms of continuity and compactness.

Findings

A space is continuous iff it is a retract of an algebraic space.

Core compactness in directed spaces relates to product equality in DTop.

Continuity and algebraicity correspond to properties of the lattice of closed sets.

Abstract

We investigate two approximation relations on a T0 topological space, the n-approximation, and the d-approximation, which are generalizations of the way-below relation on a dcpo. Different kinds of continuous spaces are defined by the two approximations and are all shown to be directed spaces. We show that the continuity of a directed space is very similar to the continuity of a dcpo in many aspects, which indicates that the notion of directed spaces is a suitable topological extension of dcpos.The main results are: (1) A topological space is continuous iff it is a retract of an algebraic space;(2) a directed space X is core compact iff for any directed space Y, the topological product is equal to the categorical product in DTop of X and Y respectively;(3) a directed space is continuous (resp., algebraic, quasicontinuous, quasialgebraic) iff the lattice of its closed subsets is continuous (resp., algebraic, quasicontinuous, quasialgebraic).