Extensions of dualities and a new approach to the Fedorchuk duality

Dimov, Georgi; Ivanova-Dimova, Elza; Tholen, Walter

doi:10.48550/arxiv.1808.06168

Cited by 1 publication

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“…Denote by CBool the category of complete Boolean algebras and Boolean homomorphisms. The following assertions, proved in [19], are very important in this paper.…”

Section: The De Vries Duality Revisitedmentioning

confidence: 83%

“…In [19] we noted that B is a coreflective subcategory of C if, and only if, there exists a class P of C-morphisms satisfying properties (P1-4) and the following strengthening of (P5 • ):…”

Section: 3mentioning

confidence: 99%

“…Lemma 4.7. ( [19]) Let A ∈ |CBool|, X ∈ |CHaus|, π : S a (A) −→ X be an irreducible mapping and for every a, b ∈ A, define…”

Section: The De Vries Duality Revisitedmentioning

confidence: 99%

“…In [19] we presented a set of axioms on the class P and a construction of the category D, which allowed us to reproduce the Fedorchuk duality [25] from the general setting. The same construction will, however, not work for the principal target of this paper, the de Vries duality [15].…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 gives the categorical extension result, as indicated in the previous paragraph. In Section 4 we give a review of mostly known facts concerning the de Vries duality, augmented by some results from [19] on which we rely heavily in our exposition. In Section 5, we derive the de Vries Duality Theorem from our general categorical Theorem 3.1.…”

Section: Introductionmentioning

confidence: 99%

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Extensions of dualities and a new approach to the Fedorchuk duality

Dimov

Ivanova-Dimova

Tholen

2020

Topology and its Applications

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We prove a general categorical theorem for the extension of dualities. Applying it, we present new proofs of the de Vries Duality Theorem for the category CHaus of compact Hausdorff spaces and continuous maps, and of the recent Bezhanishvili-Morandi-Olberding Duality Theorem which extends the de Vries duality to the category Tych of Tychonoff spaces and continuous maps. In the process of doing so we obtain new duality theorems for the categories CHaus and Tych. spaces X and Y is encoded by the Boolean homomorphisms between the Boolean algebras CO(Y ) and CO(X). It is natural to ask whether a similar result holds for all compact Hausdorff spaces and continuous maps between them. The first candidate for the role of the Boolean algebra CO(X) under such an extension seems to be the Boolean algebra RC(X) of all regular closed subsets of a compact Hausdorff space X (or, its isomorphic copy RO(X), which collects all regular open subsets of X), but it fails immediately since, as is well-known, RC(X) is isomorphic to RC(EX), where EX is the absolute of X. However, in 1962, de Vries [15] showed that, if we regard the Boolean algebra RC(X) together with the relation ρ X on RC(X), defined bythen the pair (RC(X), ρ X ) determines uniquely (up to homeomorphism) the compact Hausdorff space X. Moreover, with the help of some special maps between (RC(X), ρ X ) and (RC(Y ), ρ Y ), where X and Y are compact Hausdorff spaces, one can reconstruct all continuous maps between Y and X. De Vries gave an algebraic description of the pairs (RC(X), ρ X ) as pairs (A, C), formed by a complete Boolean algebra A and a relation C on A, satisfying some axioms, and he also described algebraically the needed special maps of such pairs. In this way he obtained the category DeV and its dual equivalence with the category CHaus of compact Hausdorff spaces and continuous maps. In fact, de Vries did not use the relation ρ X as mentioned above, but its "dual", that is, the relation F ≪ X G, defined by (F ≪ X G ⇔ F (−ρ X )G * ) (with −ρ X complementary to ρ X and G * denoting the Boolean negation of G in RC(X)) and called the non-tangential inclusion; equivalently, F ≪ X G ⇔ F ⊆ int X (G). Now known as de Vries algebras, he originally called the abstract pairs (A, ≪) compingent algebras. The axioms for the relation C (respectively, ≪ C ) on A are precisely the axioms for Efremovič proximities [23], with only one exception: instead of Efremovič's separation axiom, which refers to the points of the space in question, de Vries introduced what is now called the extensionality axiom (see [20, Lemma 2.2, p.215] for a motivation for this terminology). Since Efremovič proximities are relations on the Boolean algebra (P(X), ⊆) of all subsets of a set X, de Vries algebras may be regarded as point-free generalizations of the Efremovič proximities.Nowadays the pairs (A, C), where A is a Boolean algebra and C is a proximitytype relation on A, attract the attention not only of topologists, but also of logicians and theoretical computer sciencists. Amongst the many g...

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“…Denote by CBool the category of complete Boolean algebras and Boolean homomorphisms. The following assertions, proved in [19], are very important in this paper.…”

Section: The De Vries Duality Revisitedmentioning

confidence: 83%

Section: 3mentioning

confidence: 99%

“…Lemma 4.7. ( [19]) Let A ∈ |CBool|, X ∈ |CHaus|, π : S a (A) −→ X be an irreducible mapping and for every a, b ∈ A, define…”

Section: The De Vries Duality Revisitedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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