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CHOICE-FREE STONE DUALITY
- Source :
- Journal of Symbolic Logic, 85(1), 109-148. Cambridge University Press
- Publication Year :
- 2019
- Publisher :
- Cambridge University Press (CUP), 2019.
-
Abstract
- The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.<br />Postprint with minor updates described in footnote on page 1
- Subjects :
- Pure mathematics
Closed set
Logic
Specialization (pre)order
010102 general mathematics
Open set
Mathematics::General Topology
Mathematics - Category Theory
Mathematics - Logic
0102 computer and information sciences
Topological space
Stone duality
03G05, 06E15, 06D22, 03E25
01 natural sciences
Philosophy
Boolean prime ideal theorem
010201 computation theory & mathematics
Clopen set
FOS: Mathematics
F.4.1
Category Theory (math.CT)
Field of sets
0101 mathematics
Logic (math.LO)
Mathematics
Subjects
Details
- ISSN :
- 19435886 and 00224812
- Volume :
- 85
- Database :
- OpenAIRE
- Journal :
- The Journal of Symbolic Logic
- Accession number :
- edsair.doi.dedup.....6bc49cf4b36084841240a121f0e22cc3