On classes of T0 spaces admitting completions

Giuli, Eraldo

doi:10.4995/agt.2003.2016

Cited by 13 publications

(15 citation statements)

References 15 publications

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“…[10]) whose objects (called affine sets) are pairs ðX ; UÞ where X is a set and U any subset of the powerset PðXÞ, and whose morphisms (called affine maps) from ðX ; UÞ to ðY ; VÞ are functions X ! f Y such that f À1 ðV Þ 2 U for every V 2 V. The Zariski closure of a subset (subobject) M of the (underlying) set X of ðX ; UÞ is defined as:…”

Section: We Recall From [4] That a Subcategory B Ofmentioning

confidence: 99%

U-Closure Operators and Compactness

Castellini

Giuli

2005

Appl Categor Struct

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A notion of compactness with respect to a previously introduced notion of functor induced closure operator is presented and analyzed. Even though this new notion shows very similar properties to compactness with respect to the classical notion of categorical closure operator, in general the two concepts are different. Examples are provided. (2000): 18A22, 18A32, 06A15. Mathematics Subject Classifications

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Section: We Recall From [4] That a Subcategory B Ofmentioning

confidence: 99%

U-Closure Operators and Compactness

Castellini

Giuli

2005

Appl Categor Struct

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“…TOP as well as CL are fully embedded in the construct SSET of affine spaces and affine maps which is a host for many other subconstructs that are important to topologists [13,14] (see next section for the exact definitions). In this paper we investigate the problem of cartesian closedness for SSET and we describe the exponential objects and deduce results for its subconstructs.…”

Section: Introductionmentioning

confidence: 99%

“…In the final section of the paper we describe the cartesian closed topological hull of SSET. Remark that, as was observed by E. Giuli [13], our definition of affine spaces and maps, as we recall it in the next section, only differs slightly from the normal Boolean Chu spaces and continuous maps, as introduced by V. Pratt to model concurrent computation. Objects in SSET have a structure containing constants.…”

Section: Introductionmentioning

confidence: 99%

“…We will restrict ourselves to the affine spaces whose affine structure contains the constant functions 0 and 1. As in [13], the corresponding construct of affine spaces and affine maps will be denoted by SSET. In that paper it was proved that SSET is a well-fibred topological construct.…”

Section: Introductionmentioning

confidence: 99%

“…We will now recall a general method to construct hereditary bicoreflective subcategories of SSET [9], [10] , [13]. In order to define a subconstruct of SSET we put an algebra structure on S = {0, 1}.…”

Section: Introductionmentioning

confidence: 99%

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Exponentiality for the construct of affine sets

Claes

2008

Appl.Gen.Topol.

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Abstract. The topological construct SSET of affine sets over the two-point set S contains many interesting topological subconstructs such as TOP, the construct of topological spaces, and CL, the construct of closure spaces. For this category and its subconstructs cartesian closedness is studied. We first give a classification of the subconstructs of SSET according to their behaviour with respect to exponentiality. We formulate sufficient conditions implying that a subconstruct behaves similar to CL. On the other hand, we characterize a conglomerate of subconstructs with behaviour similar to TOP. Finally, we construct the cartesian closed topological hull of SSET. AMS Classification: 54A05, 54C35, 18D15

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A Topologist’s View of Chu Spaces

Giuli

Tholen

2007

Appl Categor Struct

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For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X op . Its full subcategory of M-extensive K-Chu spaces is topological over X when X is M-complete, for any morphism class M. Often this subcategory may be presented as a full coreflective subcategory of Diers' category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the * -autonomous category of M-extensive and M-coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.

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On classes of T0 spaces admitting completions

Cited by 13 publications

References 15 publications

U-Closure Operators and Compactness

U-Closure Operators and Compactness

Exponentiality for the construct of affine sets

A Topologist’s View of Chu Spaces

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