Egoroff, Σ, and Convergence Properties in Some Archimedean Vector Lattices

An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each {an}n∈N ⊆ A 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 formD(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. 1. Preliminaries. We list the numerous relevant definitions, with some commentary. All vector lattices (Riesz spaces) will be archimedean (see [16]) and all topological spaces will be Tychonoff ([6], [9]). Let A be a vector lattice. In A, for a (countable) sequence (un)n∈N in A: un ↓ 0 means un↓, i.e., u1 ≥ u2 ≥ · · · , and ∧A un = 0 (∧A is the infimum in A); 2010 Mathematics Subject Classification: 06F20, 46A40, 28A20, 54G05, 28A05, 06E15, 46E30, 54G10, 54H05.