Compact spaces with a P-base

Dow, Alan; Feng, Ziqin

doi:10.1016/j.indag.2021.04.002

Cited by 6 publications

(8 citation statements)

References 14 publications

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“…If X has an ω ω -base and finite scattered height, then X is proved in [3] to be countable, hence metrizable. It is proved in [5] that the same result also holds if X has finite scattered height and a P -base for some poset P with calibre (ω 1 , ω). The compact space ω 1 + 1 has an ω ω -base under the assumption ω 1 < b.…”

Section: (Scattered) Compact Spacesmentioning

confidence: 75%

“…Hence this is also true for a compact space with countable scattered height and an ω ω -base, which gives a positive answer to Problem 8.6.8 in [3]. It is worth mentioning that in [5] the authors prove that under the assumption ω 1 < b, any compact space with an ω ω -base is metrizable and any scattered compact space with an ω ω -base is countable. We also prove that D 2 \ ∆ is strongly dominated (see definition in Section 3) by the Bowtie space B, here D is the double arrow space.…”

Section: Introductionmentioning

confidence: 73%

“…Under the assumption ω 1 = b, ω 1 is a Tukey quotient of ω ω (see [19] for the discussion of the spectrum of ω ω ), hence ω 1 +1 has an ω ω -base. However, when ω 1 < b, ω 1 is not a Tukey quotient of ω ω , hence the space ω 1 + 1 doesn't have an ω ω -base, but it has a K(Q)-base (see [5]), where Q is the space of rational numbers.…”

Section: Preliminariesmentioning

confidence: 99%

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$P$-bases and Topological Groups

Feng¹

2020

Preprint

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A topological space X is defined to have a neighborhood P -base at any x ∈ X from some poset P if there exists a neighborhood base (Up[x]) p∈P at x such that Up[x] ⊆ U p ′ [x] for all p ≥ p ′ in P . We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a K(M )-base for some separable metric space M . This gives a positive answer to Problem 8.6.8 in [3].Let A(X) be the free Abelian topological group on X. It is shown that if Y is a retract of X such that the free Abelian topological group A(Y ) has a P -base and A(X/Y ) has a Q-base, then A(X) has a P × Q-base. Also if Y is a closed subspace of X and A(X) has a P -base, then A(X/Y ) has a P -base.It is shown that any Fréchet-Urysohn topological group with a K(M )-base for some separable metric space M is first-countable, hence metrizable. And if P is a poset with calibre (ω 1 , ω) and G is a topological group with a P -base, then any precompact subset in G is metrizable, hence G is strictly angelic. Applications in function spaces Cp(X) and C k (X) are discussed. We also give an example of a topological Boolean group of character ≤ d such that the precompact subsets are metrizable but G doesn't have an ω ω -base if ω 1 < d.

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Section: (Scattered) Compact Spacesmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 73%

Section: Preliminariesmentioning

confidence: 99%

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$P$-bases and Topological Groups

Feng¹

2020

Preprint

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“…Let P and Q be posets such that P ≤ T Q. It is straightforward to see that any space with a P -base also has a Q-base (see Proposition 2.1 in [7]). Hence, we get the following result.…”

Section: Sub-posets In 2 ω and ω ωmentioning

confidence: 99%

“…Topological spaces with an ω ω -base (i.e., a G-base) have been intensively studied in recent years (see [1], [4], [18], and [13]). The authors in [7] investigated the compact spaces with a P -base for some other posets, mainly K(M ) where M is a separable metric space. In this paper, we consider the subposets in ω ω and the relation between P -base and the strong Pytkeev * networks (see definition in Section 4).…”

Section: Introductionmentioning

confidence: 99%

Sub-posets in $ω^ω$ and the Strong Pytkeev$^\ast$ Property

Feng,

Nukala

2021

Preprint

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Tukey order are used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2 c -sized collection of subposets in 2 ω which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ω ω is a Tukey quotient of ω ω , we answer two open questions published in [12].The relation between P -base and strong Pytkeev * property is investigated. Let P be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a P -base has the strong Pytkeev * property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a P -base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given.

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Directed sets of topology: Tukey representation and rejection

Feng,

Gartside

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Compact spaces with a P-base

Cited by 6 publications

References 14 publications

$P$-bases and Topological Groups

$P$-bases and Topological Groups

Sub-posets in $ω^ω$ and the Strong Pytkeev$^\ast$ Property

Directed sets of topology: Tukey representation and rejection

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