PP-2018-13:
Bezhanishvili, Nick and Holliday, Wesley H.
(2018)
*Choice-free Stone duality.*
[Pre-print]

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## 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 topo-

logical 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. This connection makes clear the relation between our

point-set topological approach to choice-free Stone duality, which may be called the

hyperspace approach, and a point-free approach to choice-free Stone duality using

Stone locales. Unlike Stone's representation of Boolean algebras via Stone spaces,

our choice-free topological representation of Boolean algebras does not show that

every Boolean algebra can be represented as a �eld of sets; but like Stone's repre-

sentation, it provides the bene�t of a topological perspective on Boolean algebras,

only now without choice. In addition to representation, we establish a choice-free

dual equivalence between the category of Boolean algebras with Boolean homomor-

phisms 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.

Item Type: | Pre-print |
---|---|

Report Nr: | PP-2018-13 |

Series Name: | Prepublication (PP) Series |

Year: | 2018 |

Subjects: | Logic Mathematics |

Depositing User: | Nick Bezhanishvili |

Date Deposited: | 09 Sep 2018 11:21 |

Last Modified: | 09 Sep 2018 11:21 |

URI: | https://eprints.illc.uva.nl/id/eprint/1626 |

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