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Egoroff, σ, and convergence properties in some archimedean vector lattices

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An archimedean vector lattice A might have the following properties:
(1) the sigma property (σ): For each ${aₙ}_{n∈ ℕ} conA⁺$ there are ${λₙ}_{n ∈ ℕ} ⊆ (0,∞)$ and a ∈ A with λₙaₙ ≤ a for each n;
(2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u.
The conjunction of these two is called strongly Egoroff.
We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent to this property of X: (E) the intersection of any sequence of dense cozero-sets contains another. (In case X is zero-dimensional, (E) holds iff the clopen algebra clopX of X is a 'Egoroff Boolean algebra'.)
A crucial part of the proof is this theorem about any compact X: For any countable intersection of dense cozero-sets U, there is uₙ ↓ 0 in C(X) with {x ∈ X: uₙ(x) ↓ 0} = U. Then, we make a construction of many new X with (E) (thus, dually, strongly Egoroff D(X)), which can be F-spaces, connected, or zero-dimensional, depending on the input to the construction. This results in many new Egoroff Boolean algebras which are also weakly countably complete.
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  • Department of Mathematics, Wesleyan University, Middletown, CT 06459, U.S.A.
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  • KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm8363-2-2016
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