Atomless Parts of Spaces.

Isbell, John

doi:10.7146/math.scand.a-11409

Cited by 330 publications

(141 citation statements)

References 8 publications

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“…In [16] Isbell introduced uniformities on frames, as the precise translation into frame terms of Tukey's notion, later developed in detail by Pultr [25]. We note that, as in the case of spaces, there are several different ways of describing uniformities on frames, such as the functional uniformities of Fletcher-Hunsaker ( [7], [8]) and the entourage unifromities of Picado [21].…”

Section: Uniform Framesmentioning

confidence: 99%

“…The above dual adjunction Oٜ between the category of frames and the category of topological spaces can be easily adapted to the uniform setting, yielding a dual adjunction between the category UFrm of uniform frames (introduced by Isbell [16], and studied in detail by Pultr [25] in terms of covers; for information about other different ways of describing them see [6,22]) and the category Unif of uniform spaces of Weil [34] and Tukey [33]. Then, denoting by F 1 and F 2 , respectively, the forgetful functors Unif → Top and UFrm → Frm forgetting the uniform structure, the diagram…”

Section: Introductionmentioning

confidence: 99%

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Uniform-type structures on lattice-valued spaces and frames

García

Mardones-Pérez

Picado

et al. 2008

Fuzzy Sets and Systems

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By introducing lattice-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L an integral commutative quantale). By showing, via an intermediate L-valued structure of uniformity, how filters of covers may describe the uniform operators of Hutton, we prove that, when restricted to Girard quantales, this general framework captures a significant class of Hutton's uniform spaces. The categories of L-valued uniform spaces and L-valued uniform frames here introduced provide (in the case L is a complete chain) the missing vertices in the commutative cube formed by the classical categories of topological and uniform spaces and their corresponding pointfree counterparts (forming the base of the cube) and the corresponding L-valued categories (forming the top of the cube).

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Section: Uniform Framesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uniform-type structures on lattice-valued spaces and frames

García

Mardones-Pérez

Picado

et al. 2008

Fuzzy Sets and Systems

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“…First we recall the notion of largest pointless sublocale introduced in [1], and described in more detail in [4]. Not all locales have largest pointless sublocales but locales induced by sober T 1 spaces do.…”

Section: The Counterexamplementioning

confidence: 99%

“…We don't need to go into any detail here since we will only be using the fact that if {B j } is a collection of bases for the locales X j , then the union of the inverse images under the maps f We will also use the following theorem from [1] (see [9] or [10] for a choice-free proof).…”

mentioning

confidence: 99%

Directed inverse limits of spatial locales

Plewe

2002

Proc. Amer. Math. Soc.

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Abstract. In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.The question "Which limits are preserved by the embedding of sober topological spaces into the category of locales?", or equivalently "When is a limit of spatial locales spatial?" seems to be rather difficult. Results concerning preservation of products (and some applications) can be found in [2,7,8]. Here we consider preservation of directed inverse limits; in particular Isbell's question whether a directed inverse limit of (quasi-)compact spatial locales is again spatial [1, p. 24]. It is easy to see that if all locales are Hausdorff (i.e. if they have closed diagonals), then this question has a positive answer because (i) compact Hausdorff locales are spatial and closed under products, (ii) limits can be constructed as equalizers of a pair of maps between products of the original objects, and (iii) equalizers of maps between Hausdorff locales are closed inclusions and compactness is inherited by closed sublocales. So only the case of non-Hausdorff locales is problematic.This note consists of two parts. In the first part we give a construction which shows that directed inverse limits of compact locales need not be spatial even if all locales are furthermore locally compact and all maps open. However, in the second part we show that if all locales are locally compact and all maps proper (a weaker property suffices), then the inverse limit is again locally compact and hence spatial; here compactness is not needed.Constructions of the limit of a directed inverse system [6] or [10]. We don't need to go into any detail here since we will only be using the fact that if {B j } is a collection of bases for the locales X j , then the union of the inverse images under the maps f * ∞j is a base for the limit. This union is a subbase for any limit. But directedness implies that it is in fact a base.

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“…The famous adjunction Ω ⊣ P t between the category Top of topological spaces and the opposite Loc of the category Frm of frames [16,17,19], known as Papert-Papert-Isbell adjunction [24], and its various generalizations in fuzzy set theory have received much attention during the last three decades [2,3,4,5,14,17,20,23,24]. Also see [4,24] for many other references not included in this paper.…”

Section: Introductionmentioning

confidence: 99%

Applications of Categorical Fixed-basis Fuzzy Topological Spaces to A-valued Spaces

Demirci¹

2013

Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology

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As an application of the dual equivalence between the category of L-spatial C-objects and the category of L-sober C-M-L-spaces, it is shown in this paper that for a fixed augmented partially ordered set A, there exists a dual equivalence between the category of A-spatial augmented partially ordered sets and the category of A-sober A-valued spaces. Then, with regard to this duality, for a fixed (Z 1 , Z 2 )-complete partially ordered set L, we establish a dual equivalence between the category of L-spatial (Z 1 , Z 2 )-complete partially ordered sets and the category of L-sober L-valued Q-spaces.

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Atomless Parts of Spaces.

Cited by 330 publications

References 8 publications

Uniform-type structures on lattice-valued spaces and frames

Uniform-type structures on lattice-valued spaces and frames

Directed inverse limits of spatial locales

Applications of Categorical Fixed-basis Fuzzy Topological Spaces to A-valued Spaces

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