On the T1 axiom and other separation properties in constructive point-free and point-set topology

In this note a T1 formal space (T1 set-generated locale) is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a T1∗ formal space, and prove that the class of points of a weakly set-presentable T1∗ formal space is a set in the...

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Bibliographic Details
Published inAnnals of pure and applied logic Vol. 161; no. 4; p. 560
Main Authors Aczel, Peter, Curi, Giovanni
Format Journal Article
LanguageEnglish
Published 2010
Subjects
Constructive
Point-free
Point-set
Separation properties
Topology
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Summary:In this note a T1 formal space (T1 set-generated locale) is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a T1∗ formal space, and prove that the class of points of a weakly set-presentable T1∗ formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties Ti∗ for constructive topological spaces (ct-spaces), strengthening separation properties discussed elsewhere. Finally we relate the Ti∗ properties for ct-spaces with corresponding properties of formal spaces.
ISSN:1873-2461
0168-0072
DOI:10.1016/j.apal.2009.03.005