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Keimel, Klaus; Lawson, Jimmie D.

doi:10.1016/j.apal.2008.06.019

Cited by 68 publications

(26 citation statements)

References 19 publications

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“…Domain theory plays a foundational role in denotational semantics of programming languages. In domain theory, the d-spaces, well-filtered spaces and sober spaces form three of the most important classes (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23][24][25][26][27][28][29]). Let Top 0 be the category of all T 0 spaces and Sob the category of all sober spaces.…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that Sob is reflective in Top 0 (see [6,9]). Using d-closures, Wyler [22] proved that Top d is reflective in Top 0 (see also [16,27]). Later, Ershov [5] showed that the d-completion of X (i.e., the d-reflection of X) can be obtained by adding the closure of directed sets onto X (and then repeating this process by transfinite induction).…”

Section: Introductionmentioning

confidence: 99%

“…Later, Ershov [5] showed that the d-completion of X (i.e., the d-reflection of X) can be obtained by adding the closure of directed sets onto X (and then repeating this process by transfinite induction). In [16], Keimel and Lawson proved that for a full subcategory K of Top 0 containing Sob, if K has certain properties, then K is reflective in Top 0 . They showed that Top d and some other categories have such properties.…”

Section: Introductionmentioning

confidence: 99%

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A direct approach to K-reflections of T0 spaces

2020

Topology and its Applications

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In this paper, we provide a direct approach to K-reflections of T 0 spaces. For a full subcategory K of the category of all T 0 spaces and a T 0 space X, let K(X) = {A ⊆ X : A is closed and for any continuousWe call K an adequate category if for any T 0 space X, P H (K(X)) is a K-space. Therefore, if K is adequate, then K is reflective in Top 0 . It is shown that the category of all sober spaces, that of all d-spaces, that of all well-filtered spaces and the Keimel and Lawson's category are all adequate, and hence are all reflective in Top 0 . Some major properties of K-spaces and K-reflections of T 0 spaces are investigated. In particular, it is proved that if K is adequate, then the K-reflection preserves finite products of T 0 spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A direct approach to K-reflections of T0 spaces

2020

Topology and its Applications

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“…Later, Ershov [5] showed that the d-completion (i.e., the d-reflection) of X can be obtained by adding the closure of directed sets onto X (and then repeating this process by transfinite induction). In [18], using Wyler's method, Keimel and Lawson proved that for a full subcategory K of Top 0 containing Sob, if K has certain properties, then K is reflective in Top 0 . They showed that Top d and some other categories have such properties.…”

Section: Introductionmentioning

confidence: 99%

On H-sober spaces and H-sobrifications of $T_0$ spaces

Xu¹

2020

Preprint

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In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. The concepts of irreducible subset systems (R-subset systems for short), H-sober spaces and super H-sober spaces for a general R-subset system H are introduced. It is proved that the product space of a family of T 0 spaces is H-sober iff each factor space is H-sober, and if H has a natural property (called property M), then the super H-sobriety is a special type of H-sobriety, and hence the product space of a family of T 0 spaces is super H-sober iff each factor space is super Hsober. Let Top 0 be the category of all T 0 spaces with continuous mappings. For a T 0 space X and an H-sober space Y , we show that the function space Top 0 (X, Y ) equipped with the topology of pointwise convergence is H-sober. Furthermore, if H has property M and Y is a super H-sober space, then the function space Top 0 (X, Y ) equipped with the topology of pointwise convergence is super H-sober. One immediate corollary is that for a T 0 space X and a well-filtered space Y , the function space Top 0 (X, Y ) equipped with the topology of pointwise convergence is well-filtered. For an R-subset system H having property M, the Smyth power space of a H-sober space is not H-sober in general. But for the super H-sobriety, we prove that a T 0 space X is super H-sober iff its Smyth power space P S (X) is super H-sober. A direct construction of the H-sobrifcations and super H-sobrifications of T 0 spaces is given. So the category of all H-sober spaces is reflective in Top 0 , and the category of all super H-sober spaces is also reflective in Top 0 if H has property M. It is shown that the H-sobrification preserves finite products of T 0 spaces, and the super H-sobrification preserves finite products of T 0 spaces if H has property M.

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“…It is well-known that the category of all sober spaces and that of d-spaces are reflective in the category of all T 0 spaces (see [7,12,23]). Recently, following Ershov's method of constructing the d-completion of T 0 spaces, Shen, Xi, Xu and Zhao [22] presented a construction of the well-filtered reflection of T 0 spaces.…”

Section: Introductionmentioning

confidence: 99%

First countability, $ω$-well-filtered spaces and reflections

Xu¹,

Shen²,

Xi³

et al. 2019

Preprint

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We first introduce and study two new classes of subsets in T 0 spaces -ω-Rudin sets and ω-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces -ω-d spaces and ω-well-filtered spaces. We prove that an ω-well-filtered T 0 space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable ω-well-filtered T 0 space are directed. Therefore, a first countable T 0 space X is sober iff X is well-filtered iff X is an ω-well-filtered d-space. Using ω-well-filtered determined sets, we present a direct construction of the ω-well-filtered reflections of T 0 spaces, and show that products of ω-well-filtered spaces are ω-well-filtered.

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D-completions and thed-topology

Cited by 68 publications

References 19 publications

A direct approach to K-reflections of T0 spaces

A direct approach to K-reflections of T0 spaces

On H-sober spaces and H-sobrifications of $T_0$ spaces

First countability, $ω$-well-filtered spaces and reflections

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