Egoroff, $\sigma $, and convergence properties in some archimedean vector lattices
Tom 231 / 2015
Streszczenie
An archimedean vector lattice $A$ might have the following properties:
(1) the sigma property ($\sigma$): For each $\{a_n\}_{n\in\mathbb N} \operatorname{con} A^+$ there are $\{\lambda_n\}_{n\in\mathbb N}\subseteq (0,\infty)$ and $a\in A$ with $\lambda_n a_n \le a$ for each $n$;
(2) order convergence and relative uniform convergence are equivalent, denoted $(\operatorname{OC} \Rightarrow$ $\operatorname{RUC})$: if $a_n \downarrow 0$ then $a_n \to 0$ 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 $(\operatorname{OC} \Rightarrow \operatorname{RUC})$ are equivalent, and equivalent to this property of $X$: $(\mathrm{E})$ the intersection of any sequence of dense cozero-sets contains another. (In case $X$ is zero-dimensional, $(\mathrm{E})$ holds iff the clopen algebra $\operatorname{clop} X$ 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 $u_n \downarrow 0$ in $C(X)$ with $ \{x\in X : u_n(x) \downarrow 0\} = U. $ Then, we make a construction of many new $X$ with $(\mathrm{E})$ (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.
