Well-filterifications of topological spaces

Liu, Bei; Li, Qingguo; Wu, Guohua

doi:10.1016/j.topol.2020.107245

Cited by 17 publications

(21 citation statements)

References 8 publications

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“…It is not difficult to check that the space X is a WD space but not a Rudin space. Therefore, Example 4.15 in [36] also gave a negative answer to another related question raised by Xu, Shen, Xi and Zhao in [51]: Is every well-filtered determined space a Rudin space?…”

Section: Rudin Sets and Well-filtered Determined Setsmentioning

confidence: 99%

“…( [51]) For any T 0 space X, S c (X) ⊆ D c (X) ⊆ RD(X) ⊆ WD(X) ⊆ Irr c (X). In [36,Example 4.15], Liu, Li and Wu constructed a T 0 space X in which some well-filtered determined sets are not Rudin sets, and hence gave a negative answer to a queston posed by Xu and Zhao in [55]: Does RD(X) = WD(X) hold for every T 0 space X? It is not difficult to check that the space X is a WD space but not a Rudin space.…”

Section: Rudin Sets and Well-filtered Determined Setsmentioning

confidence: 99%

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Some open problems on well-filtered spaces and sober spaces

Zhao

2021

Topology and its Applications

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Section: Rudin Sets and Well-filtered Determined Setsmentioning

confidence: 99%

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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“…Lemma 2.16. [9] Let N = (N, τ cof ), where τ cof denotes the cofinite topology, and X n an irreducible T 0 space for every n ∈ N , such that there are at most finitely many X n 's that have a greatest element under the specialization order…”

Section: Definition 23 ([4]mentioning

confidence: 99%

“…In [9], Liu, Li and Wu give a counterexample to show that WD(X) may not agree with KF(X) for any T 0 space X, which solved the open problem proposed by Xu in [15]. Zhang and Li provided a direct characterization of the D-completion of a T 0 space using the tapered closed subsets ( [16]).…”

Section: Introductionmentioning

confidence: 99%

D-completion, well-filterification and sobrification

Miao¹,

Wang²,

Li³

2021

Preprint

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In this paper, we obtain some sufficient conditions for the D-completion of a T 0 space to be the well-filterification of this space, the well-filterification of a T 0 space to be the sobrification of this space and the D-completion of a T 0 space to be the sobrification, respectively. Moreover, we give an example to show that a tapered closed set may be neither the closure of a directed set nor the closed KF -set, respectively. Because the tapered closed set is a closed W D-set, the example also gives a negative answer to a problem proposed by Xu. Meantime, a new direct characterization of the D-completion of a T 0 space is given by using the notion of pre-c-compact elements.

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“…In [14,Example 4.15], a well-filtered space X but not a Rudin space was given. It is straightforward to check that X is not first-countable.…”

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confidence: 99%

$ω$-Rudin spaces, well-filtered determined spaces and first-countable spaces

Xu,

Shen,

et al. 2020

Preprint

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We investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties. The main results include: (i) if the sobrification of a T 0 space X is first-countable, then X is an ω-Rudin space; (ii) every ω-well-filtered space is sober if its sobrification is first-countable; (iii) if a T 0 space is second-countable or first-countable and with a countable underlying set, then it is a ω-Rudin space; (iv) every first-countable T 0 space is well-filtered determined; (v) every irreducible closed subset in a firstcountable ω-well-filtered space is countably-directed; (vi) every first-countable ω-well-filtered ω * -d-space is sober.

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Well-filterifications of topological spaces

Cited by 17 publications

References 8 publications

Some open problems on well-filtered spaces and sober spaces

Some open problems on well-filtered spaces and sober spaces

D-completion, well-filterification and sobrification

$ω$-Rudin spaces, well-filtered determined spaces and first-countable spaces

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