Calibres, compacta and diagonals

Gartside, Paul; Morgan, Jeremiah

doi:10.4064/fm232-1-1

Cited by 4 publications

(4 citation statements)

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“…Let K be a compact space. Gartide and Morgan show in [13] that K is compact if the off-diagonal subspace K 2 \ ∆ has a compact cover D with Calibre (ω 1 , ω) which swallows all the compact sets of K 2 \ ∆, i.e., D is cofinal in K(K 2 \ ∆).…”

Section: Topological Groupsmentioning

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: Topological Groupsmentioning

confidence: 99%

“…Therefore, the collection L swallows all the compact subsets of K × K \ ∆. Then by [13], K is metrizable. This finishes the proof.…”

Section: Topological Groupsmentioning

confidence: 99%

$P$-bases and Topological Groups

Feng¹

2020

Preprint

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“…With regard to [6,Problem 4.3] specifically, we can mention the paper [12] where Gartside and Morgan proved that a compact space is metrizable whenever the complement of its diagonal in its square has caliber (ω 1 , ω, ω). Also, in [7], Guerrero and Dow proved that assuming CH a compact space X is metrizable whenever it has a P-diagonal, and later, the same conclusion was obtained in ZFC by Dow and Hart in [8].…”

Section: Introductionmentioning

confidence: 99%

Spaces with an M-diagonal

Sánchez

2019

RACSAM

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“…In particular, Cascales and Orihuela in [2] proved that any compact space X with X 2 \ ∆ being strongly P-dominated is metrizable. In [3], it was proved that a compact space X is metrizable if X 2 \ ∆ is strongly M-dominated for some separable metric space M. Recently, Gartside and Morgan in [8] proved that any compact space X is metrizable if X 2 \∆ has a cofinal P -directed compact cover for some directed set P with calibre (ω 1 , ω) (see definition in Section 2). It is not known whether the word 'strongly' or 'cofinal' can be omitted in these results.…”

Section: Introductionmentioning

confidence: 99%