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Title: Axiom TD and the Simmons sublocale theorem
Authors: Picado, Jorge 
Pultr, Aleš
Keywords: Frame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space.
Issue Date: 2019
Publisher: Mathematical Institute of Charles University
Project: UID/MAT/00324/2019 
Serial title, monograph or event: Comment.Math.Univ.Carolin.
Volume: 60
Issue: 4
Abstract: More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem.
DOI: 10.14712/1213-7243.2019.030
Rights: embargoedAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais

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