Gδ-diagonals and metrization theorems

McArthur, William G.

doi:10.2140/pjm.1973.44.613

Cited by 21 publications

(7 citation statements)

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“…For example, every Moore space has a G δ -diagonal of rank 2 [2], while not every Moore space has a regular G δ -diagonal [17]. Also, by a theorem of McArthur [20] every pseudocompact space with a regular G δ -diagonal is compact, while in [2] it is pointed out that the Mrowka space, which is a pseudocompact and not countably compact space, has a G δ -diagonal of rank 2. (Indeed, if X = ω ∪ A, where A is a MAD family in ω, by letting…”

Section: Cardinal Inequalities For Spaces Withmentioning

confidence: 99%

On cardinality bounds involving the weak Lindelöf degree

Bella

Carlson

2017

Quaestiones Mathematicae

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We give a general closing-off argument in Theorem 2.1 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2 wL(X)ψ(X) , and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2 wL(X)t(X) . The first extends the well-known cardinality bound 2 ψ(X) for a compactum X in a new direction. As |X| ≤ 2 wL(X)χ(X) for a normal space X [3], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.10 we give a short, direct proof of (1) that does not use 2.1. Yet 2.1 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base B such that |B| ≤ 2 wL(X)χ(X) for all B ∈ B, then |X| ≤ 2 wL(X)χ(X) .Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2 wL(X)ψ(X)t(X) . This partially answers a question from [3] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2 ℵ0 , then |X| ≤ 2 ℵ0 .Result (2) above is a new generalization of the cardinality bound 2 t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [2], De la Vega in the homogeneous case [9]). To this end we show that if U ⊆ clD ⊆ X, where X is power homogeneous and U is open, then |U | ≤ |D| πχ(X) . This is a strengthening of a result of Ridderbos [18].2010 Mathematics Subject Classification. 54D20, 54D45, 54A25, 54E99.

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Section: Cardinal Inequalities For Spaces Withmentioning

confidence: 99%

On cardinality bounds involving the weak Lindelöf degree

Bella

Carlson

2017

Quaestiones Mathematicae

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“…Let a compactum K and a point p e X -K be given. For each x e K, there exists an integer n(x) and open sets U(x) and V(x) containing p and x, respectively, In [18], it is proved that a completely regular pseudocompact space with a regular (τ δ -diagonal is metrizable. Even though the condition that the space have a regular G δ -diagonal cannot be weakened to c-stratifiability (see 3.2) as shown in 6.6, we are able to prove the following.…”

Section: (1) a Regular Space Is Nagata If And Only If It Is A C-stratmentioning

confidence: 99%

Spaces in which compacta are uniformly regularG_δ

Lee¹

1979

Pacific J. Math.

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“…Symmetrizable spaces are D-spaces (D. Burke [9]); therefore, (8) takes care of (9). For (10), it suffices to refer to Theorem 1.13 and to a result of W. McArthur in [17]: every pseudocompact Tychonoff space with a regular G δ -diagonal is metrizable. This covers (9) as well.…”

Section: Introduction and Elementary Factsmentioning

confidence: 99%

Locally compact spaces of countable core and Alexandroff compactification

Arhangel'skiı̆

2007

Topology and its Applications

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We introduce a new cardinal invariant, core of a space, defined for any locally compact Hausdorff space X and denoted by cor(X). Locally compact spaces of countable core generalize locally compact σ -compact spaces in a way that is slightly exotic, but still quite natural. We show in Section 1 that under a broad range of conditions locally compact spaces of countable core must be σ -compact. In particular, normal locally compact spaces of countable core and realcompact locally compact spaces of countable core are σ -compact. Perfect mappings preserve the class of spaces of countable core in both directions (Section 2). The Alexandroff compactification aX is weakly first countable at the Alexandroff point a if and only if cor(X) = ω (Section 3). Two examples of non-σ -compact locally compact spaces of countable core are discussed in Section 3. We also extend the well-known theorem of Alexandroff and Urysohn on the cardinality of perfectly normal compacta to compacta satisfying a weak version of perfect normality. Several open problems are formulated.

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G_δ-diagonals and metrization theorems

Cited by 21 publications

References 5 publications

On cardinality bounds involving the weak Lindelöf degree

On cardinality bounds involving the weak Lindelöf degree

Spaces in which compacta are uniformly regularG_δ

Locally compact spaces of countable core and Alexandroff compactification

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Gδ-diagonals and metrization theorems

Cited by 21 publications

References 5 publications

On cardinality bounds involving the weak Lindelöf degree

On cardinality bounds involving the weak Lindelöf degree

Spaces in which compacta are uniformly regularGδ

Locally compact spaces of countable core and Alexandroff compactification

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Product

Resources

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G_δ-diagonals and metrization theorems

Spaces in which compacta are uniformly regularG_δ