Axiom T D and the Simmons sublocale theorem

Jorge Picado; Aleš Pultr

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 4, page 541-551
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Picado, Jorge, and Pultr, Aleš. "Axiom $T_D$ and the Simmons sublocale theorem." Commentationes Mathematicae Universitatis Carolinae 60.4 (2019): 541-551. <http://eudml.org/doc/295062>.

@article{Picado2019,
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.},
author = {Picado, Jorge, Pultr, Aleš},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {frame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space},
language = {eng},
number = {4},
pages = {541-551},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Axiom $T_D$ and the Simmons sublocale theorem},
url = {http://eudml.org/doc/295062},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Picado, Jorge
AU - Pultr, Aleš
TI - Axiom $T_D$ and the Simmons sublocale theorem
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 4
SP - 541
EP - 551
AB - 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.
LA - eng
KW - frame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space
UR - http://eudml.org/doc/295062
ER -

References

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  8. Picado J., Pultr A., Frames and Locales, Topology without Points, Frontiers in Mathematics, 28, Springer, Basel, 2012. MR2868166
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