First-countability, ω-Rudin spaces and well-filtered determined spaces

Xu, Xiaoquan; Shen, Chong; Xi, Xiaoyong; Zhao, Dongsheng

doi:10.1016/j.topol.2021.107775

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…In Section 5, we recall some concepts and results about the topological Rudin Lemma, Rudin spaces, ω-Rudin spaces, well-filtered determined spaces and ω-well-filtered determined spaces in [10,24,29,30,31] that will be used in the next four sections.…”

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

confidence: 99%

“…From Theorem 5.10 and Theorem 5.23 we immediately deduce the following result. It is still not known whether a first-countable T 0 space is a Rudin space (see [31,Problem 6.15]). Since the first-countability is a hereditary property, from Remark 2.4 and Theorem 5.23 we know that if the Smyth power space P S (X) of a T 0 space X is first-countable, then X is a WD space.…”

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

confidence: 99%

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

confidence: 99%

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

confidence: 99%

On Scott power spaces

Xu¹,

Wen²,

Xi³

2022

Preprint

Self Cite

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“…In [24, Theorem 6.10 and Theorem 6.15] it was proved that every locally compact (resp., core-compact) T 0 space is a RD (resp., WD) space. It was also shown in [25,Theorem 5.6 and Theorem 6.12] that every T 0 space with a first-countable sobrification is an RD space and every first-countable T 0 space is a WD space.…”

Section: Introductionmentioning

confidence: 99%

“…In domain theory and non-Hausdorff topology, the sober spaces, well-filtered spaces and d-spaces form three of the most important classes (see [2][3][4][5][6][7][8][9][10][11][12][13][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]). Let Top 0 be the category of all T 0 spaces with continuous mappings and Sob the full subcategory of Top 0 containing all sober spaces.…”

Section: Introductionmentioning

confidence: 99%

Non-reflective categories of some kinds of weakly sober spaces

Wen

2022

Topology and its Applications

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Scott topology on Smyth power posets

Xu¹,

Wen²,

Xi³

2023

Math. Struct. Comp. Sci.

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For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$ . $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$ ) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$ . A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated.

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First-countability, ω-Rudin spaces and well-filtered determined spaces

Cited by 6 publications

References 19 publications

On Scott power spaces

On Scott power spaces

Non-reflective categories of some kinds of weakly sober spaces

Scott topology on Smyth power posets

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