Asymmetry and bicompletion of approach spaces

Brümmer, G. C. L.; Sioen, Mark

doi:10.1016/j.topol.2005.06.015

Cited by 6 publications

(11 citation statements)

References 14 publications

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“…The previous result solves the open problem stated in [3] on the characterization of the epis in AP 0 .…”

Section: (Weak) Hereditariness Of the Closure Operatorsupporting

confidence: 63%

“…In order to describe r on M C ξ some symmetrization is involved. The definition below is inspired by the one given in [3] in the setting of approach spaces. …”

Section: (Weak) Hereditariness Of the Closure Operatormentioning

confidence: 99%

“…In some subconstructs of AP 0 a solution of the epimorphism problem is known [10], however in AP 0 itself the problem of characterizing the epimorphisms is still open. It was stated as an open problem in [3].…”

Section: Introductionmentioning

confidence: 99%

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Epimorphisms and cowellpoweredness for separated metrically generated theories

2007

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For metrically generated constructs X we give an internal characterization of the regular closure operator on X, determined by the subconstruct X 0 , consisting of its T 0 objects. This allows us to describe the epimorphisms in X 0 and to show that all the constructs of that type are cowellpowered. We capture many known results but our method also gives solutions in cases where the epimorphism problem was still open.

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“…The previous result solves the open problem stated in [3] on the characterization of the epis in AP 0 .…”

Section: (Weak) Hereditariness Of the Closure Operatorsupporting

confidence: 63%

“…In order to describe r on M C ξ some symmetrization is involved. The definition below is inspired by the one given in [3] in the setting of approach spaces. …”

Section: (Weak) Hereditariness Of the Closure Operatormentioning

confidence: 99%

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Epimorphisms and cowellpoweredness for separated metrically generated theories

2007

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“…There is an easy criterion to see whether for a reflector R, a class of morphisms U satisfying (1), (2), and (3) is the largest one with these properties. A morphism class U is said to be coessential if for every morphism u ∈ U and for every morphism f we have u • f ∈ U ⇒ f ∈ U.…”

Section: Preliminariesmentioning

confidence: 99%

“…A more precise definition is formulated in the next preliminary section. Two standard examples are the following: The usual bicompletion functor for quasi uniform spaces with associated firm class of morphisms consisting of all epimorphic embeddings and the bicompletion functor for approach spaces [2] for which the associated firm class coincides with the class of all embeddings that are dense with respect to the quasi metric coreflection. In this paper we prove that these basic constructions carry over to arbitrary metrically generated constructs.…”

Section: Introductionmentioning

confidence: 99%

Bicompletion of metrically generated constructs

Colebunders

Vandersmissen

2010

Acta Math Hung

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We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms L(R) is known. This functor will permit to "lift" the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory R F for the construct X0 of T0 objects and that the associated firm class of morphisms can be derived from L(R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones.

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Uniform structures in the beginning of the third millenium

Künzi

2007

Topology and its Applications

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Asymmetry and bicompletion of approach spaces

Cited by 6 publications

References 14 publications

Epimorphisms and cowellpoweredness for separated metrically generated theories

Epimorphisms and cowellpoweredness for separated metrically generated theories

Bicompletion of metrically generated constructs

Uniform structures in the beginning of the third millenium

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