Cardinality bounds via covers by compact sets

Bella, Angelo; Carlson, Nathan

doi:10.1007/s10474-020-01103-9

Cited by 3 publications

(8 citation statements)

References 17 publications

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“…This improves the result of Arhangel ′ skiȋ, van Mill, and Ridderbos in [2] that |X| ≤ 2 t(X) for a power homogeneous compactum X and gives a partial answer to a question in [4]. In addition, if X is a homogeneous Hausdorff space we show that |X| ≤ 2 pwc L(X)wt(X)πχ(X)pct(X) , improving a result in [3]. It also extends the result in [4] into the Hausdorff setting.…”

supporting

confidence: 83%

“…We show in Theorem 4.4 that if X is additionally homogeneous then |X| ≤ 2 pwLc(X)wt(X)πχ(X)pct(X) . This improves Corollary 3.7 in [3], which states that |X| ≤ 2 pwLc(X)t(X)pct(X) if X is homogeneous and Hausdorff. Theorem 4.4 also generalizes Theorem 1.1(b) into the class of Hausdorff spaces.…”

Section: Introductionsupporting

confidence: 69%

“…Despite the fact that four cardinal invariants are used in the bound in Theorem 4.4, each of these invariants should be regarded as "small" in a rough sense. In addition, as πχ(X) ≤ t(X)pct(X) for a Hausdorff space X and wt(X) ≤ t(X), we see that Theorem 4.4 improves Corollary 3.7 in [3]. This latter result states that if X is homogeneous and Hausdorff, then |X| ≤ 2 pwLc(X)t(X)pct(X) .…”

Section: Definition 42 ([5]mentioning

confidence: 61%

“…The following improves Theorem 3.5 in [3] by replacing t(X) with wt(X). Unlike Theorem 3.5 in [3], the closing-off argument uses an increasing chain of C-saturated sets.…”

Section: Definition 42 ([5]mentioning

confidence: 92%

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Power homogeneous compacta and variations on tightness

Carlson¹

2021

Preprint

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The weak tightness wt(X), introduced in [6], has the property wt(X) ≤ t(X). It was shown in [4] that if X is a homogeneous compactum then |X| ≤ 2 wt(X)πχ(X) . We introduce the almost tightness at(X) with the property wt(X) ≤ at(X) ≤ t(X) and show that if X is a power homogeneous compactum then |X| ≤ 2 at(X)πχ(X) . This improves the result of Arhangel ′ skiȋ, van Mill, and Ridderbos in [2] that |X| ≤ 2 t(X) for a power homogeneous compactum X and gives a partial answer to a question in [4]. In addition, if X is a homogeneous Hausdorff space we show that |X| ≤ 2 pwc L(X)wt(X)πχ(X)pct(X) , improving a result in [3]. It also extends the result in [4] into the Hausdorff setting. The cardinal invariant pwLc(X), introduced in [5] by Bella and Spadaro, satisfies pwLc(X) ≤ L(X) and pwLc(X) ≤ c(X). We also show the weight w(X) of a homogeneous space X is bounded in various contexts using wt(X). One such result is that if X is homogeneous and regular then w(X) ≤ 2 L(X)wt(X)pct(X) . This generalizes a result in [4] that if X is a homogeneous compactum then w(X) ≤ 2 wt(X) .

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supporting

confidence: 83%

Section: Introductionsupporting

confidence: 69%

Section: Definition 42 ([5]mentioning

confidence: 61%

“…The following improves Theorem 3.5 in [3] by replacing t(X) with wt(X). Unlike Theorem 3.5 in [3], the closing-off argument uses an increasing chain of C-saturated sets.…”

Section: Definition 42 ([5]mentioning

confidence: 92%

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Power homogeneous compacta and variations on tightness

Carlson¹

2021

Preprint

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“…Recently in [13], the Lindelöf degree L(X) in Theorem 6.1 was replaced by the cardinal invariant pwL c (X), introduced by Bella and Spadaro in [16]. The piecewise weak Lindelöf degree for closed sets pwL c (X) of X is the least infinite cardinal κ such that for every closed set F ⊆ X, for every open cover U of F , and every decomposition {U i : i ∈ I} of U, there are families…”

Section: Generalizations Of De La Vega's Theoremmentioning

confidence: 99%

A survey of cardinality bounds on homogeneous topological spaces

Carlson¹

2020

Preprint

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In this survey we catalogue the many results of the past several decades concerning bounds on the cardinality of a topological space with homogeneous or homogeneous-like properties. These results include van Douwen's Theorem, which states |X| ≤ 2 πw(X) if X is a power homogeneous Hausdorff space [25], and its improvements |X| ≤ d(X) πχ(X) [42] and |X| ≤ 2 c(X)πχ(X)[18] for spaces X with the same properties. We also discuss de la Vega's Theorem, which states that |X| ≤ 2 t(X) if X is a homogeneous compactum [24], as well as its recent improvements and generalizations to other settings. This reference document also includes a table of strongest known cardinality bounds on spaces with homogeneous-like properties. The author has chosen to give some proofs if they exhibit typical or fundamental proof techniques. Finally, a few new results are given, notably (1) |X| ≤ d(X) πnχ(X) if X is homogeneous and Hausdorff, and (2) |X| ≤ πχ(X) c(X)qψ(X) if X is a regular homogeneous space. The invariant πnχ(X), defined in this paper, has the property πnχ(X) ≤ πχ(X) and thus (1) improves the bound d(X) πχ(X) for homogeneous Hausdorff spaces. The invariant qψ(X), defined in [32], has the properties qψ(X) ≤ πχ(X) and qψ(X) ≤ ψc(X) if X is Hausdorff, thus (2) improves the bound 2 c(X)πχ(X) in the regular, homogeneous setting.

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Power homogeneous compacta and variations on tightness

Carlson

2023

Topology and its Applications

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Cardinality bounds via covers by compact sets

Cited by 3 publications

References 17 publications

Power homogeneous compacta and variations on tightness

Power homogeneous compacta and variations on tightness

A survey of cardinality bounds on homogeneous topological spaces

Power homogeneous compacta and variations on tightness

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