A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. I

Dimov, Georgi

doi:10.1007/s10474-010-0035-5

Cited by 20 publications

(70 citation statements)

References 26 publications

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As proved in [8], there exists a duality Λ t between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight w a and of dimension dim a of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X) = w a (Λ t (X)), and if, in addition, X is normal, then dim(X) = dim a (Λ t (X)).

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mentioning

confidence: 89%

“…(We do not give here the explicit definition of the contravariant functor Λ a because we will not use it. (It is given in [8].) For our purposes here, it is enough to know that the compositions Λ a • Λ t and Λ t • Λ a are naturally equivalent to the corresponding identity functors (see, e.g., [1]). )…”

Section: Local Contact Algebrasmentioning

confidence: 99%

“…Subsequently, Dimov [8] extended de Vries' duality from the category HC to the category HLC of locally compact Hausdorff spaces and continuous maps, and, on the base of this result, he also obtained an extension of Stone's duality from the category Stone of compact zero-dimensional Hausdorff spaces and continuous maps to the category ZHLC of zero-dimensional locally compact Hausdorff spaces and continuous maps (see [10,7]).…”

Section: Introductionmentioning

confidence: 99%

“…The paper [8] has its precursor in results by P. Roeper [25], who showed that all information about a locally compact Hausdorff space X is contained in the triple RC(X), ρ X , CR(X) , where CR(X) is the set of all compact regular closed subsets of X. In order to describe abstractly the triples RC(X), ρ X , CR(X) , he introduced the notion of region-based topology, and he proved that -up to homeomorphisms, respectively, isomorphisms -there exists a bijection between the class of all locally compact Hausdorff spaces and the class of all complete region-based topologies.…”

Section: Introductionmentioning

confidence: 99%

“…In order to describe abstractly the triples RC(X), ρ X , CR(X) , he introduced the notion of region-based topology, and he proved that -up to homeomorphisms, respectively, isomorphisms -there exists a bijection between the class of all locally compact Hausdorff spaces and the class of all complete region-based topologies. The duality theorem proved in [8] says that there exists a duality Λ t between the category HLC and the category DHLC of all complete region-based topologies and appropriate morphisms between them. Note that Λ t (X) df = RC(X), ρ X , CR(X) , for every locally compact Hausdorff space X.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On dimension andweight of a local contact algebra

Dimov¹,

Ivanova-Dimova²,

Düntsch³

2018

Filomat

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“…

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mentioning

confidence: 89%

Section: Local Contact Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On dimension andweight of a local contact algebra

Dimov¹,

Ivanova-Dimova²,

Düntsch³

2018

Filomat

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A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. II

Dimov

2010

Acta Math Hung

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This paper is a continuation of [7], where a duality theorem for the category HLC of locally compact Hausdorff spaces and continuous maps is proved. In the present paper, we characterize the injective and surjective morphisms of the category HLC, as well as of its cofull subcategories determined by the open, by the skeletal or by the perfect maps, by means of some corresponding properties of their dual morphisms. This is in analogy with some well-known results of M. H. Stone [17] and generalizes some similar results of de Vries [4]. Such characterizations are also obtained for the homeomorphic embeddings, dense embeddings, LCA-embeddings etc.; in particular a theorem of Fedorchuk [10, Theorem 6] is generalized. Again in analogy with some well-known results of M. H. Stone [17], the dual objects of the open, regular open, clopen, compact open, regular closed etc.subsets of a locally compact Hausdorff space X are directly described by means of the dual object of X; some of these results are new even in the compact case. An explicit description of the products of local contact algebras (= LC-algebras) in the category DHLC dual to the category HLC is given and a completion theory for LC-algebras is developed.

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Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

Bezhanishvili

Sourabh

et al. 2016

Appl Categor Struct

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By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative "modal-like" duality by introducing the concept of a Gleason space, which is a pair (X, R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.

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A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. I

Abstract: Generalizing de Vries' duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.

Cited by 20 publications

References 26 publications

On dimension andweight of a local contact algebra

On dimension andweight of a local contact algebra

A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. II

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

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