Directed complete poset models of T1 spaces

Abstract

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space P is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.