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

An archimedean vector lattice A might have the following properties: (1) the sigma property (sigma): For each {a(n)}(n) is an element of N subset of A(+) there are {lambda(n)}(n) is an element of N subset of (0, infinity) and a is an element of A with lambda(n)a(n) <= a for each 12; (2) order convergence and relative uniform convergence are equivalent, denoted (OC double right arrow RUC): if a(n) down arrow 0 then a(n) -> 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 (sigma) and (OC double right arrow 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 clop X 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 u(n) down arrow 0 in C(X) with {x is an element of X : u(n)(x) down arrow 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.