Open AccessOn complete objects in the category of T0 closure spaces
Author(s) -
Didier Deses,
Eraldo Giuli,
E. LowenColebunders
Publication year - 2003
Publication title -
applied general topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.638
H-Index - 13
eISSN - 1989-4147
pISSN - 1576-9402
DOI - 10.4995/agt.2003.2007
Subject(s) - mathematics , closure (psychology) , injective function , characterization (materials science) , completeness (order theory) , class (philosophy) , closure operator , combinatorics , joins , sobriety , discrete mathematics , complete lattice , topological space , pure mathematics , closed set , mathematical analysis , artificial intelligence , computer science , psychology , materials science , physics , universality (dynamical systems) , quantum mechanics , economics , market economy , psychotherapist , programming language , nanotechnology
In this paper we present an example in the setting of closure spaces that fits in the general theory on “complete objects” as developed by G. C. L. Brümmer and E. Giuli. For V the class of epimorphic embeddings in the construct Cl0 of T0 closure spaces we prove that the class of V-injective objects is the unique firmly V-reflective subconstruct of Cl0. We present an internal characterization of the Vinjective objects as “complete” ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. An external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. We prove that the construct of complete T0 closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins
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