Egoroff, $\sigma $, and convergence properties in some archimedean vector lattices

A. W. Hager ¹   ; J. van Mill ²

¹ Department of Mathematics Wesleyan University Middletown, CT 06459, U.S.A.
² KdV Institute for Mathematics University of Amsterdam Science Park 105-107 P.O. Box 94248 1090 GE Amsterdam, The Netherlands

Studia Mathematica, Tome 231 (2015) no. 3, pp. 269-285

Résumé

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.

Détail

DOI : 10.4064/sm8363-2-2016

Keywords: archimedean vector lattice might have following properties sigma property sigma each mathbb operatorname con there lambda mathbb subseteq infty lambda each nbsp order convergence relative uniform convergence equivalent denoted operatorname rightarrow operatorname ruc downarrow conjunction these called strongly egoroff consider vector lattices form extended real continuous functions compact space showing sigma operatorname rightarrow operatorname ruc equivalent equivalent property mathrm intersection sequence dense cozero sets contains another zero dimensional mathrm holds clopen algebra operatorname clop egoroff boolean algebra crucial part proof theorem about compact countable intersection dense cozero sets there downarrow downarrow make construction many mathrm dually strongly egoroff which f spaces connected zero dimensional depending input construction results many egoroff boolean algebras which weakly countably complete