- Directed complete poset models of T1 spaces

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# Directed complete poset models of T1 spaces

Citation

Zhao, D., & Xi, X. (2018). Directed complete poset models of T1 spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 164(1), 125-134. https://doi.org/10.1017/S0305004116000888

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.

Date Issued

2018

DOI

10.1017/S0305004116000888