Hausdorff separation in categories
type: Item
instance of: scholarly article
title: Hausdorff separation in categories
zbMATH Open document ID: 0796.18005
DOI: 10.1007/BF00873991
published in: Applied Categorical Structures
publication date: 1994-09-25T00:00:00Z
review text: The basis for the investigations in this paper consists of a category \({\mathcal X}\), a class \({\mathcal M}\) of morphisms in \({\mathcal X}\) closed under composition and containing all isomorphisms, and \({\mathcal X}\) is assumed to be \({\mathcal M}\)-complete. This implies that there is a unique class of morphisms \({\mathcal E}\subset {\mathcal X}\), such that \(({\mathcal E},{\mathcal M})\) is a factorization system. As an additional parameter a class \({\mathcal P}\) of \({\mathcal X}\)-objects closed under morphisms of \({\mathcal E}\) is introduced. For any object \(X\) the class of points \(pt(X)\) consists of those subobjects of \(X\) with domain in \({\mathcal P}\). As a third parameter a closure operator \(C\) with respect to \({\mathcal M}\), as defined by \textit{D. Dikranjan} and \textit{E. Giuli} [Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)], is considered in \({\mathcal X}\). With respect to these three parameters \({\mathcal M}\), \({\mathcal P}\) and \(C\), \(T_ 2\)-objects in \({\mathcal X}\) are defined and the full subcategory of \(T_ 2\)-objects is studied. Under a smallness condition imposed on \({\mathcal P}\), more precisely on \(pt(X)\) for any object \(X\), one gets a canonical functor \(Pt\) from the category of \(T_ 2\)-objects to the category of closure spaces. Under natural conditions this functor preserves and reflects \(T_ 2\)-objects, which implies that the category of \(T_ 2\)-objects of \({\mathcal X}\) is closed under monosources. In the last section, connections between the subcategory of \(T_ 2\)-objects and the subcategory of separated objects of \({\mathcal X}\), defined by \(C\)-closed diagonals are investigated. The introduction of the additional parameter \({\mathcal P}\) allows a wider range of applications and a finer analysis of the notion \(T_ 2\) than former investigations, which used the absolute notion of ``constant''.
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 18A32
zbMATH DE Number: 554354
zbMATH Keywords: separated object
zbMATH Keywords: constant morphism with respect to a class
zbMATH Keywords: factorization system
zbMATH Keywords: closure operator
zbMATH Keywords: \(T_ 2\)-objects
zbMATH Keywords: closure spaces
author: Maria Manuel Clementino
reviewed by: Dieter Pumplün
MaRDI profile type: Publication
cites work: Q3715851
cites work: Closure operators. I
cites work: Q5609457
cites work: A diagonal theorem for epireflective subcategories of Top and cowellpoweredness
cites work: Objects with closed diagonals
cites work: Factorizations, denseness, separation, and relatively compact objects
cites work: Compact Hausdorff objects
cites work: Separated totally convex spaces
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cites work: Q5605131
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description: scientific article; zbMATH DE number 554354