Sober approach spaces

Banaschewski, Bernhard; Löwen, R.; Olmen, Christophe Van

doi:10.1016/j.topol.2006.05.003

Cited by 20 publications

(44 citation statements)

References 3 publications

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“…An approach frame L, as introduced in [4], is a frame (with top and bottom ⊥) equipped with two families of unary operations on L, addition and subtraction with α ∈ [0, ∞], denoted A α and S α respectively and which are required to satisfy all the identities valid for addition and subtraction by α, and the frame operations in [0, ∞]. Notably the following hold:…”

Section: 3mentioning

confidence: 99%

“…Now let X be an approach space and consider its regular function frame RX. As we know [4], this is an approach frame and hence we can reformulate the above denitions in this context. Then RX is regular if for all θ < ∞, f ∈ RX, ε > 0 and x ∈ X there exist g, h ∈ RX such that g f , g ∧ h = 0, f ∨ (h − θ) ∈ N RX and f (x) g(x) + θ + ε, and analogously for pre-regular with θ = 0.…”

Section: Regularity In Approach Framesmentioning

confidence: 99%

“…Now, in the rst place, in topology regularity is characterized using closed sets and in approach theory the role of the collection of closed sets is played by the so-called regular function frame. In the second place, in [4] it was proved that it is possible to eliminate from the regular function frame the role of the points of the underlying set in the same way as was done in the transition from topology to frames. Taking as starting point the original denitions of notions of regularity in approach spaces [11] given by means of lters, we prove that it is possible not only to give a characterization using the regular function frame but more generally to formulate this in the setting of approach frames in such a way that it generalises at the same time the usual concepts for topological spaces, approach spaces and frames, while at the same time maintaining important stability properties.…”

Section: Introductionmentioning

confidence: 99%

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Regularity in approach theory

2007

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The setting of approach frames is an exquisite environment to compare the various notions of regularity in topology, frames and approach spaces and in this paper we make this investigation. This brings to light some interesting similarities and dierences and proves that there are at least two dierent unifying notions in the setting of approach frames each with specic advantages. * The third author is an Aspirant of the Foundation for Scientic Research Flanders.

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Section: 3mentioning

confidence: 99%

Section: Regularity In Approach Framesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularity in approach theory

2007

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“…all continuous maps from X to the Sierpinski space on {0, 1}. It was also developed for a sober approach space X and the frame of all contractions to some Sierpinski object on [0, ∞], [4]. In both cases, it follows from the existing duality that sober spaces form an epireective subcategory of the subconstruct Top 0 (resp.…”

Section: Introductionmentioning

confidence: 99%

“…Ap 0 ) of T 0 objects. The actual epireection of a T 0 topological space (approach space), which is also called the sobrication, can then be constructed either using the duality [17,4] or using the Sierpinski object [16,15]. In this paper, we construct the sobrication of a T 0 approach space via bicompletion of a particular associated quasi-uniform gauge structure.…”

Section: Introductionmentioning

confidence: 99%

Constructing the sobrification of an approach space via bicompletion

Gerlo

Vandersmissen

2007

Acta Math Hung

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On every approach space X, we construct a compatible quasiuniform gauge structure which turns out to be at the same time the coarsest functorial structure and the nest compatible totally bounded one. Based on the analogy with the classical CsászárPervin quasi-uniform space, we call this the CsászárPervin quasi-uniform gauge space. By means of the bicompletion of this CsászárPervin quasi-uniform gauge space of a T0 approach space X, we succeed in constructing the sobrication of X.

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Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

Gerlo¹,

Vandersmissen²,

Olmen³

2006

Appl Categor Struct

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In [2] the subconstruct Sob of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category Ap 0 of T 0 approach spaces. The main result of this paper states that moreover Sob is firmly U-reflective in Ap 0 for the class U of epimorphic embeddings. 'Firm U-reflective' is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category Unif 0 of Hausdorff uniform spaces with uniformly continuous maps. It means that Sob is U-reflective in Ap 0 and that the reflector is such that f : X → Y belongs to U if and only if ( f ) is an isomorphism. Firm U-reflectiveness implies uniqueness of completion in the sense that whenever f : X → Y is a map with f ∈ U and Y sober, the associated f * : (X ) → Y is an isomorphism. Our result generalizes the fact that in the category Top 0 the subconstruct of sober topological spaces is firmly reflective for the class U b of b-dense embeddings in Top 0 . Also firmness in some other subconstructs of Ap 0 will be easily obtained.

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Sober approach spaces

Cited by 20 publications

References 3 publications

Regularity in approach theory

Regularity in approach theory

Constructing the sobrification of an approach space via bicompletion

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

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