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Zhang, Xiao‐Dong; Schaefer, Helmut; Winkowska-Nowak, Katarzyna

doi:10.1023/a:1009712030403

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2024

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“…The restriction of τ to C x being a compact FN -topology, C x is atomic. This is a consequence of a result of I. Kaplansky of 1947; see [12,Satz 4.6, (2)⇒( 6)] or [13, Theorem 6.3, (4)⇒( 3)]; see also [15,Theorem 2.4, (1)⇒( 3)]. (Some comments on the final part of the proof of [13,Theorem 6.3] are given in Remark 5 below.)…”

Section: And So C /mentioning

confidence: 81%

Compactness of order intervals in a locally solid linear lattice

Lipecki

2022

Colloq. Math.

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Let X be a linear lattice, let x ∈ X+, and let τ be a locally solid topology on X. We present four conditions equivalent to the τ -compactness of the order interval [0, x] in X, including the following ones: (i) there is a set S and an affine homeomorphism of [0, x] onto the Tychonoff cube [0, 1] S which preserves order; (ii) Cx, the set of components of x, is τ -compact and [0, x] is order σ-complete. In the special case where X is a Banach lattice and τ is its norm topology, another equivalent condition is: (iii) Cx is weakly compact and [0, x] is order σ-complete.

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Section: And So C /mentioning

confidence: 81%

Compactness of order intervals in a locally solid linear lattice

Lipecki

2022

Colloq. Math.

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Atomicity of Boolean algebras and vector lattices in terms of order convergence

Avilés,

Bilokopytov,

Troitsky

2024

Proc. Amer. Math. Soc.

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We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.

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Cited by 2 publications

References 9 publications

Compactness of order intervals in a locally solid linear lattice

Compactness of order intervals in a locally solid linear lattice

Atomicity of Boolean algebras and vector lattices in terms of order convergence

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