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arXiv · 2609.29498

Aliprantis's questions on locally solid topologies

David Muñoz-Lahoz·Mitchell A. Taylor·Pedro Tradacete
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Abstract

In 1974, C. D. Aliprantis posed several questions concerning topological completions of locally solid vector lattices. We answer all of them in the negative. We construct Hausdorff locally convex-solid vector lattices showing that, without metrizability, neither the $σ$-Lebesgue property nor property (B, i) need pass to the completion; a positive element of the completion need not be the limit of a decreasing sequence of upper elements; the generalized (A, 0) property need not make the canonical image a regular sublattice; and regularity of this image need not imply order density. All five counterexamples have been formalized in Lean 4 using the Banach lattice Lean library.

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David Muñoz-Lahoz, Mitchell A. Taylor, Pedro Tradacete. 2026-08-24. Aliprantis's questions on locally solid topologies. https://arxiv.org/abs/2609.29498

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