Categories of Locally Hypercompact Spaces and Quasicontinuous Posets

Erné, Marcel

doi:10.1007/s10485-018-9536-0

Cited by 33 publications

(28 citation statements)

References 20 publications

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“…For {F i : i ∈ I} ⊆ P (<ω) and x ∈ P , we show that ↓ i∈I (↑x ∩ ↓F i ) ∈ C(ΣP ). For any D ∈ D(P ) with (2). It follows that ↑d ⊆ P \ ↓(↑x ∩ i∈I ↓F i ) for some d ∈ D, and whence ↑d ∩ ↑x ⊆ P \ i∈I ↓F i , proving that (P, υ(P )) is a strong d-space.…”

Section: Some New Characterizations Of Well-filtered Spaces and Sobermentioning

confidence: 90%

On topological Rudin's lemma, well-filtered spaces and sober spaces

Zhao

2020

Topology and its Applications

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Based on topological Rudin's Lemma, we investigate two new kinds of sets -Rudin sets and well-filtered determined sets in T 0 topological spaces. Using such sets, we formulate and prove some new characterizations for well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on wellfiltered spaces. Our study also lead to a new class of spaces -strong d-spaces and some problems whose solutions will strengthen our understanding of the related structures.

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Section: Some New Characterizations Of Well-filtered Spaces and Sobermentioning

confidence: 90%

On topological Rudin's lemma, well-filtered spaces and sober spaces

Zhao

2020

Topology and its Applications

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“…As in [13], a topological space X is locally hypercompact if for each x ∈ X and each open neighborhood…”

Section: Preliminarymentioning

confidence: 99%

“…Lemma 5.11. ( [13]) Let X be a locally hypercompact T 0 space and A ∈ Irr(X). Then there exists a directed subset D ⊆ ↓A such that A = D.…”

Section: Rudin Sets and Well-filtered Determined Setsmentioning

confidence: 99%

Some open problems on well-filtered spaces and sober spaces

Zhao

2021

Topology and its Applications

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“…As in [4], a topological space X is locally hypercompact if for each x ∈ X and each open neighborhood U of x, there is ↑F ∈ Fin X with x ∈ int ↑F ⊆ ↑F ⊆ U . A space X is called core compact if O(X) is a continuous lattice (see [6]).…”

Section: Preliminarymentioning

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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Categories of Locally Hypercompact Spaces and Quasicontinuous Posets

Cited by 33 publications

References 20 publications

On topological Rudin's lemma, well-filtered spaces and sober spaces

On topological Rudin's lemma, well-filtered spaces and sober spaces

Some open problems on well-filtered spaces and sober spaces

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

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