Finite approximation of stably compact spaces

Abstract

Finite approximation of spaces by inverse sequences of graphs (in the category of so-called topological graphs) was introduced by Smyth, and developed further. The idea was subsequently taken up by Kopperman and Wilson, who developed their own purely topological approach using inverse spectra of finite T₀-spaces in the category of stably compact spaces. Both approaches are, however, restricted to the approximation of (compact) Hausdorff spaces and therefore cannot accommodate, for example, the upper space and (multi-) function space constructions. We present a new method of finite approximation of stably compact spaces using finite stably compact graphs, which when the topology is discrete are simply finite directed graphs. As an extended example, illustrating the problems involved, we study (ordered spaces and) arcs.