On complete objects in the category of T0 closure spaces


  • D. Deses Vrije Universiteit Brussel
  • Eraldo Giuli Università di L'Aquila
  • E. Lowen-Colebunders Vrije Universiteit Brussel




Complete object, Firm, Injective, Complete lattice, T0 object, Closure space


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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Author Biographies

D. Deses, Vrije Universiteit Brussel

Department of Mathematics

Eraldo Giuli, Università di L'Aquila

Dipartimento di Matematica Pura ed Applicata

E. Lowen-Colebunders, Vrije Universiteit Brussel

Department of Mathematics


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How to Cite

D. Deses, E. Giuli, and E. Lowen-Colebunders, “On complete objects in the category of T0 closure spaces”, Appl. Gen. Topol., vol. 4, no. 1, pp. 25–34, Apr. 2003.



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