Productivity of Zariski-compactness for constructs of affine spaces

Claes, Veerle; Lowen-Colebunders, E.

doi:10.1016/j.topol.2005.01.007

Cited by 5 publications

(3 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will look for cartesian closed topological constructs that are larger than SSET and in which SSET is finally densely embedded. We know from [7] that quotients in SSET are productive. In fact we have the same situation as for the category CL [6].…”

Section: Cartesian Closed Topological Hull Of Ssetmentioning

confidence: 99%

Exponentiality for the construct of affine sets

Claes

2008

Appl.Gen.Topol.

View full text Add to dashboard Cite

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

show abstract

Section: Cartesian Closed Topological Hull Of Ssetmentioning

confidence: 99%

Exponentiality for the construct of affine sets

Claes

2008

Appl.Gen.Topol.

View full text Add to dashboard Cite

show abstract

“…We do not have productiveness of b -compactness. For instance in [3], it was stated that infinite powers of the Sierpinski space S 2 are not b -compact. Note that the same example can be used to point out that D * -and P * -compactness are not productive in Ap.…”

Section: Examplesmentioning

confidence: 99%

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

Gerlo

2007

Appl Categor Struct

View full text Add to dashboard Cite

In a topological construct X endowed with a proper (E, M)-factorization system and a concrete functor : X → Prap, we study F -compactness and FHausdorff separation, where F is a class of "closed morphisms" in the sense of Clementino et al. (A functional approach to general topology. In: Categorical by . In particular, we point out under which conditions on , the notion of F -compactness of an object X of X coincides with 0-compactness of the image (X) in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b -compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of bcompactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability.

show abstract

“…Veerle Claes and Eva Colebunders [6] found sufficient conditions for the equality z-compact = complete showing in particular that the equality holds in Sep SET{0, 1} and in Sep CS.…”

Section: Proposition 12 a Is Z-compactmentioning

confidence: 99%

Zariski closure, completeness and compactness

Giuli

2006

Topology and its Applications

View full text Add to dashboard Cite

The categorical theory of closure operators is used to introduce and study separated, complete and compact objects with respect to the Zariski closure operator naturally defined in any category X (A, Ω) obtained by a given complete category X (endowed with a proper factorization structure for morphisms) and by a given X -algebra (A, Ω) by forming the affine X -objects modelled by (A, Ω).Several basic examples are provided.

show abstract

Productivity of Zariski-compactness for constructs of affine spaces

Cited by 5 publications

References 9 publications

Exponentiality for the construct of affine sets

Exponentiality for the construct of affine sets

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

Zariski closure, completeness and compactness

Contact Info

Product

Resources

About