Cardinality restrictions on power homogeneous T5 compacta

Ridderbos, G.J.F.

doi:10.1556/sscmath.2008.1079

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…Lemma 4.1 shows that the existence of a dense set of points of small π-character in a power homogeneous space guarantees that every point in the space has small π-character. Lemma 4.1 (Ridderbos, [17]). Let X be a power homogeneous space and let κ be an infinite cardinal.…”

Section: Power Homogeneous Compactamentioning

confidence: 99%

On the weak tightness and power homogeneous compacta

Carlson

2018

Topology and its Applications

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Section: Power Homogeneous Compactamentioning

confidence: 99%

On the weak tightness and power homogeneous compacta

Carlson

2018

Topology and its Applications

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“…It follows from Theorem 4.1 that the cardinality of a homogeneous T 5 compactum is at most 2 c(X) . This was observed by van Mill in [38] and was proved for power homogeneous T 5 compacta in [43]. Note additionally that Corollary 5.15 states that the cardinality of a homogeneous T 5 compactum is at most 2 wt(X) .…”

Section: Questions and A Table Of Boundsmentioning

confidence: 52%

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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“…Seeking a contradiction, suppose X , D, and κ are as in the previous corollary. By Proposition 2.1 of [20], 2 χ (Y ) 2 πχ(Y )c(Y ) for every power homogeneous compactum Y . Hence, by GCH, κ πχ(X)c(X).…”

Section: Applications To Power Homogeneous Compactamentioning

confidence: 98%

“…• |X| 2 t( X) for every power homogeneous compactum X [2]. • |X| 2 c( X) for every T 5 homogeneous compactum X [20].…”

Section: Lemma 15 ([15 Lemma 24]) Every Preordered Set P Has a Comentioning

confidence: 99%

Power homogeneous compacta and the order theory of local bases

Milovich

Ridderbos

2011

Topology and its Applications

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We show that if a power homogeneous compactum X has character κ + and density at most κ, then there is a nonempty open U ⊆ X such that every p in U is flat, "flat" meaning that p has a family F of χ (p, X)-many neighborhoods such that p is not in the interior of the intersection of any infinite subfamily of F . The binary notion of a point being flat or not flat is refined by a cardinal function, the local Noetherian type, which is in turn refined by the κ-wide splitting numbers, a new family of cardinal functions we introduce. We show that the flatness of p and the κ-wide splitting numbers of p are invariant with respect to passing from p in X to p α<λ in X λ , provided that λ < χ (p, X), or, respectively, that λ < cf κ. The above < χ (p, X)-power-invariance is not generally true for the local Noetherian type of p, as shown by a counterexample where χ (p, X) is singular.

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Cardinality restrictions on power homogeneous T5 compacta

Cited by 4 publications

References 7 publications

On the weak tightness and power homogeneous compacta

On the weak tightness and power homogeneous compacta

A survey of cardinality bounds on homogeneous topological spaces

Power homogeneous compacta and the order theory of local bases

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