Separation versus connectedness
type: Item
instance of: scholarly article
title: Separation versus connectedness
zbMATH Open document ID: 0906.18003
DOI: 10.1016/S0166-8641(96)00087-9
author: Maria Manuel Clementino
author: Walter Tholen
published in: Topology and its Applications
publication date: 1998-12-13T00:00:00Z
review text: For a category \({\mathcal X}\) with finite products and a closure operator \(C\) an object \(X\) is \(C\)-separated (\(C\)-connected) if the diagonal \(\Delta_X\) in \(X\times X\) is \(C\)-closed (\(C\)-dense, respectively). Here closure operator is intended in the sense of Giuli and the reviewer [cf. \textit{E. Giuli} and \textit{D. Dikranyan}, Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)] where these notions where introduced. While the subcategory \(\Delta(C)\) of \(C\)-separated objects is better understood and described via the diagonal theorem [cf. \textit{E. Giuli} and \textit{H. Hušek}, Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006) and \textit{E. Giuli, S. Mantovani} and \textit{W. Tholen}, J. Pure Appl. Algebra 51, No. 1/2, 129-140 (1988; Zbl 0651.18002)] the subcategory \(\nabla(C)\) of \(C\)-connected objects was studied only recently in the monograph [\textit{D. Dikranyan} and \textit{W. Tholen}, ``Categorical structure of closure operators. With applications to topology, algebra and discrete mathematics'', Mathematics and its Applications 346, Kluwer Academic Publishers, Dordrecht (1995; Zbl 0853.18002)]. The present authors obtain a stronger form of the diagonal theorem and introduce the important notion of coregular closure operator that permits to describe similarly the subcategory \(\nabla(C)\) of \(C\)-connected objects. The pairs disconnectedness/connectedness are considered often as a pair of left constant and right constant categories. The authors contribute essentially also in this direction by proving that the category \(\nabla(C)\) is left constant iff \(C\) is regular, while the category \(\Delta(C)\) is right constant iff \(C\) is coregular. The authors give a very elegant diagram (Union Jack) of four commutative triangles of Galois correspondences involving closure operators of a category \({\mathcal X}\), subcategories of \({\mathcal X}\) and morphisms of \({\mathcal X}\). Only one of these triangles was known pr…
reviewed by: Dikran Dikranjan
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 18E40
zbMATH DE Number: 988060
zbMATH Keywords: separation
zbMATH Keywords: closure operator
zbMATH Keywords: connectedness
zbMATH Keywords: four commutative triangles of Galois correspondences
zbMATH Keywords: factorization
zbMATH Keywords: Pumplün-Röhrl connection
MaRDI profile type: Publication
cites work: Q4693062
cites work: Connectednesses and disconnectednesses in topology
cites work: Multicoreflective subcategories and coprime objects
cites work: Q4353288
cites work: Closure operators and connectedness
cites work: A factorization of the Pumplün-Röhrl connection
cites work: Weakly hereditary regular closure operators
cites work: Q4371032
cites work: Constant morphisms and constant subcategories
cites work: Q3130029
cites work: A Torsion Theory for Abelian Categories
cites work: Q3715851
cites work: Closure operators. I
cites work: Q4859897
cites work: Erratum
cites work: A diagonal theorem for epireflective subcategories of Top and cowellpoweredness
cites work: Objects with closed diagonals
cites work: Topologische Reflexionen und Coreflexionen
cites work: DISCONNECTEDNESS
cites work: Some Remarks Concerning Categories and Subspaces
cites work: Monomorphisms, Epimorphisms, and Pull-Backs
cites work: Q3932046
cites work: On two extension of dicksons torsion theory
cites work: Trennung und Zusammenhang
cites work: Separated totally convex spaces
cites work: Diagonalisierungspaare. I
cites work: Q4102451
cites work: Q5663081
cites work: Q4045297
cites work: Semi-topological functors. I
cites work: Q3338413
cites work: Q4353294
cites work: COMPONENT SUBCATEGORIES
cites work: On the classes of dense and closed subobjects
full work available at URL: s0166-8641(96)00087-9
OpenAlex ID: W2003794953
description: scientific article; zbMATH DE number 988060