On complete objects in the category of T0 closure spaces

Abstract

Abstract. 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. 2000

Epimorphisms and cowellpoweredness for separated metrically generated theories

“…So far only a few partial results are known in the literature. For instance for the construct of Birkhoff closure spaces isomorphic to (M C ∆ η T ) 0 it is known that the closure u coincides with the b-closure [6], and therefore it is hereditary. No results are known to the authors with respect to ( M C ∆ η U ) 0 .…”

Zariski closure, completeness and compactness

“…We obtain the category CS of closure spaces. The complete closure spaces are the ones in which every closed set is the closure of an unique (uniqueness ensures separation) point [8].…”

“…In [8] the complete objects in the category of T 0 closure are characterized internally (see example above) and externally.…”

On classes of T0 spaces admitting completions

“…For CS an internal characterization of the complete ( = absolutely closed) T 0 spaces is given in [9]; in Top 0 they are the sober T 0 spaces (see the introduction) and in PrOS it is easy to see that they are the partially ordered sets.…”