Dcpo models of T1 spaces

Zhao Dongsheng; Xi Xiaoyong

doi:10.29007/prcv

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Dcpo models of T1 spaces

4 pages•Published: July 28, 2014

Dongsheng Zhao and Xi Xiaoyong

Abstract

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

Keyphrases: Continuous poset, dcpo, poset model of topological space, sober space

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 221--224

Links:	https://easychair.org/publications/paper/2J1
	https://doi.org/10.29007/prcv

BibTeX entry

@inproceedings{TACL2013:Dcpo_models_of_Tsub1_sub,
  author    = {Zhao Dongsheng and Xi Xiaoyong},
  title     = {Dcpo models of  T \textbackslash{}ensuremath\{ \ensuremath{< }  \} sub \textbackslash{}ensuremath\{ \ensuremath{> }  \} 1 \textbackslash{}ensuremath\{ \ensuremath{< }  \} /sub \textbackslash{}ensuremath\{ \ensuremath{> }  \}    spaces},
  booktitle = {TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic},
  editor    = {Nikolaos Galatos and Alexander Kurz and Constantine Tsinakis},
  series    = {EPiC Series in Computing},
  volume    = {25},
  pages     = {221--224},
  year      = {2014},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {https://easychair.org/publications/paper/2J1},
  doi       = {10.29007/prcv}}

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Dcpo models of T<sub>1</sub> spaces

Abstract