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Directed complete poset models of T1 spaces

Published online by Cambridge University Press:  11 October 2016

DONGSHENG ZHAO
Affiliation:
Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore637616. e-mail: dongshengzhao@nie.edu.sg
XIAOYONG XI
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, China221116. e-mail: littlebrook@jsnu.edu.cn

Abstract

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP 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 Σ E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

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