Gap functionals, proximities and hyperspace compactification

Maio, Giuseppe Di; Löwen, R.; Naimpally, S. A.; Sioen, Mark

doi:10.1016/j.topol.2005.01.021

Cited by 9 publications

(8 citation statements)

References 10 publications

(12 reference statements)

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“…Application of ξ U G to M C for C = C ∆ , C = C θ and C = C ∆,s yields isomorphic descriptions of the construct qUG of quasi-uniform gauge spaces [16], the construct qEfGap of quasi Efremovic gap spaces [10] and the construct UG of uniform gauge spaces [16]. These are in some sense quantified versions of the corresponding constructs defined by ξ T and ξ U .…”

Section: Preliminariesmentioning

confidence: 99%

“…Particular examples of metrically generated theories to which our completion theory is applicable, are the well known constructs qUnif, qProx, Top, consisting of all quasi uniform spaces, all quasi proximity spaces, all topological spaces respectively, and all of their quantitative counterparts, like for instance the construct qUG of quasi uniform gauge spaces [16], qEfGap of all quasi Efremovic gap spaces [10], or Ap the construct of approach spaces [15]. When we apply our completion technique to symmetric structures, which was the more restrictive setting of [7], we recover the list of examples discussed in detail in that paper.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bicompletion of metrically generated constructs

Colebunders

Vandersmissen

2010

Acta Math Hung

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“…Considering the expanders on other categories M C gives the possibility of describing many other examples: for C equal to C ∆s , the construct M C ξ is isomorphic to UNIF in case ξ equals ξ C U , to the construct CREG of completely regular topological spaces in case ξ equals ξ C T , to the construct UG of uniform gauge spaces [5] in case ξ equals ξ C U G , to UAP in case ξ equals ξ C A and to C ∆s in case ξ equals ξ C D . In [5], also totally bounded metrics were considered: for C equal to C ∆ϑ and for ξ equal to ξ U the construct M C ξ is isomorphic to QPROX, for C equal to C ∆sϑ and again with ξ U the construct M C ξ is isomorphic to PROX and for C equal to C ∆sϑ and with ξ equal to ξ U G the construct M C ξ is isomorphic to efGAP [9].…”

Section: Acta Mathematica Hungarica 114 2007mentioning

confidence: 99%

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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“…When applied to M C , for C consisting of all totally bounded metric spaces, the coreflective subconstructs M C ξ are isomorphic to Creg, UAp, Prox (proximity spaces), Gap (Efgap spaces [8]) and to the construct of totally bounded metric spaces respectively.…”

Section: Concrete Examples Of Uniform Completenessmentioning

confidence: 99%

Uniqueness of completion for metrically generated constructs

Colebunders

Löwen

Vandersmissen

2007

Topology and its Applications

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In this paper, for metrically generated constructs X in the sense of [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] we study completion as a U -reflector R on the subconstruct X 0 of all T 0 -objects, for U some class of embeddings. Roughly speaking we deal with constructs X that are generated by the subclass of their metrizable objects and for various types of completion functors R available in that context, we obtain internal descriptions of the largest class U for which completion is unique. We apply our results to some well known situations. Completion of uniform spaces, of proximity spaces or of non-Archimedian uniform spaces is unique with respect to the class of all epimorphic embeddings, and this class is the largest one. However the largest class of morphisms for which Dieudonné completion of completely regular spaces or of zero dimensional spaces is unique, is strictly smaller than the class of all epimorphic embeddings. The same is true for completion in quantitative theories like uniform approach spaces for which the largest U coincides with the class of all embeddings that are dense with respect to the metric coreflection. Our results on completion for metrically generated constructs explain these differences.

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Gap functionals, proximities and hyperspace compactification

Cited by 9 publications

References 10 publications

Bicompletion of metrically generated constructs

Bicompletion of metrically generated constructs

Epimorphisms and cowellpoweredness for separated metrically generated theories

Uniqueness of completion for metrically generated constructs

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