Some open problems on well-filtered spaces and sober spaces

Xu, Xiaoquan; Zhao, Dongsheng

doi:10.1016/j.topol.2020.107540

Cited by 12 publications

(5 citation statements)

References 53 publications

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“…The main purpose of this paper is to make similar considerations, along the aforementioned development by Ershov in [2], in the context of another important class of topological spaces -the well-filtered spaces -which is attracting an increasing amount of attention in recent years ( [12,13,[15][16][17][18][19]). So, what are well-filtered spaces?…”

Section: Introductionmentioning

confidence: 92%

The wf-rank of topological spaces

Liu

2022

Filomat

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Section: Introductionmentioning

confidence: 92%

The wf-rank of topological spaces

Liu

2022

Filomat

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“…Question 3.7. ( [47]) Is there a complete Boolean algebra B such that Σ B is not sober? In 2019, in a talk [18] given at the National Institute of Education, Singapore, Achim Jung also asked whether there is a countable complete lattice whose Scott space is non-sober.…”

Section: Well-filtered Spaces and Sober Spacesmentioning

confidence: 99%

Non-Hausdorff topology

Xu¹

2023

Topology - Recent Advances and Applications [Working Title]

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In most topology books, the Hausdorff separation property is assumed from the very start and contain very little information on non-Hausdorff spaces. In classical mathematics, most topological spaces are indeed Hausdorff. But non-Hausdorff spaces are important already in algebraic geometry, and crucial in fields such as domain theory. Indeed, in connection with order, non-Hausdorff spaces, especially $T_0$ spaces, play a more significant role than Hausdorff spaces. Sobriety is probably the most important and useful property of non-Hausdorff topological spaces. It has been used in the characterizations of spectral spaces and $T_0$ spaces that are determined by their open set lattices. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology theory: $d$-space and well-filtered space. In the past few years, some remarkable progresses have been achieved in understanding such structures. In this chapter, we shall make a brief survey on some of these progresses and list a few related problems.

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“…It is still not known whether every core-compact T 0 space is a Rudin space (see [34,Question 5.14]).…”

Section: Topological Rudin Lemma Rudin Spaces and Well-filtered Deter...mentioning

confidence: 99%

On Scott power spaces

Xu¹,

Wen²,

Xi³

2022

Preprint

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In this paper, we mainly discuss some basic properties of Scott power spaces. For a T 0 space X, let K(X) be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order. It is proved that the Scott power space ΣK(X) of a well-filtered space X is still well-filtered, and a T 0 space Y is well-filtered iff ΣK(Y ) is well-filtered and the upper Vietoris topology is coarser than the Scott topology on K(Y ). A sober space is constructed for which its Scott power space is not sober. A few sufficient conditions are given under which a Scott power space is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces and Scott power spaces are also investigated.

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Some open problems on well-filtered spaces and sober spaces

Cited by 12 publications

References 53 publications

The wf-rank of topological spaces

The wf-rank of topological spaces

Non-Hausdorff topology

On Scott power spaces

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